text:
\begin{document}

\title[Patterson-Sullivan measures of point processes]{Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions}

\author
{Alexander I. Bufetov}
\address
{Alexander I. BUFETOV:
Aix-Marseille Universit\'e, Centrale Marseille, CNRS, Institut de Math\'ematiques de Marseille, UMR7373, 39 Rue F. Joliot Curie 13453, Marseille, France;
Steklov Mathematical Institute of RAS, Moscow, Russia}
\email{bufetov@mi.ras.ru, alexander.bufetov@univ-amu.fr}

\author
{Yanqi Qiu}
\address
{Yanqi QIU: Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, China; CNRS, Institut de Math\'ematiques de Toulouse, Universit\'e Paul Sabatier
}

\email{yanqi.qiu@amss.ac.cn}

\begin{abstract}
The Patterson-Sullivan construction is proved almost surely to recover every Hardy function from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the conformal invariance of the point process and the slow growth of variance for linear statistics. Patterson-Sullivan reconstruction of Hardy functions is obtained in real and complex hyperbolic spaces of arbitrary dimension, while reconstruction of continuous functions is established in general $\mathrm{CAT}(-1)$ spaces.
\end{abstract}
\dedicatory{To the memory of Alexander Ivanovich Balabanov (1952 -- 2018)}
\subjclass[2010]{Primary 60G55; Secondary 37D40, 32A36}
\keywords{Patterson-Sullivan measure, point processes, negative correlations, determinantal measures, Poisson processes,
Bergman space, Hardy space, conformal invariance}

\maketitle

\tableofcontents

\setcounter{equation}{0}

\section{Introduction}\label{sec-intro}

\subsection{The key example}\label{sec-Z-f}

Let $(g_n)_{n\ge 0}$ be a sequence of independent complex Gaussian random variables with expectation $0$ and variance $1$. The random series $\mathfrak{f}_\D(z) = \sum_{n=0}^\infty g_n z^n$ almost surely defines a holomorphic function on the unit disk $\D$.  Peres and Vir\'ag \cite{PV-acta} proved  that the law of the zero set
of $\mathfrak{f}_\D$  is the determinantal point process induced by the Bergman kernel $K_{\D}$ on the disk.
Let ${\Pi}_{K_{\D}}$ be the corresponding measure  on the space $\Conf(\D)$ of configurations on $\D$.  For ${\Pi}_{K_{\D}}$-almost any $X\in\Conf(\D)$, it is
proved in \cite{BQS-LP}  that   a holomorphic function on the unit disk, square-integrable with respect to the Lebesgue measure, and equal to $0$ in restriction
to $X$, must be the zero function; in other words,  $X$ is a  {\it uniqueness set} for the Bergman space $A^2(\D)\subset L^2(\D, dV)$ of  holomorphic functions on $\D$ square-integrable with respect to the Lebesgue measure $dV$.

 \begin{question}\label{ques-motiv}
How to recover a  function $f \in A^2(\D)$ from its restriction to  $X$?
\end{question}

Erasing a finite number of points from an $A^2(\D)$-uniqueness set still yields an $A^2(\D)$-uniquess set, therefore, the reconstruction algorithm that we are looking for in Question \ref{ques-motiv} should not depend on any finite part of our set but should reflect its asymptotic behaviour at the boundary.  The following preliminary proposition shows that for {\it fixed} $f \in A^2(\D)$ and {\it fixed} $z \in \D$, one can reconstruct $f(z)$ from the restriction $f|_X$ onto
 ${\Pi}_{K_{\D}}$-almost every $X\in\Conf(\D)$.
  The argument proceeds by adapting the Patterson-Sullivan construction, see Patterson \cite{Patterson-acta} and Sullivan \cite{Sullivan-IHES}.
Set
\begin{equation} \label{lob-dist}
d_\D(x, z) := \log  \frac{{\displaystyle 1 +  \Big|\frac{z-x}{1 - \bar{x}z}\Big|}}{{\displaystyle 1 - \Big|\frac{z-x}{1 - \bar{x}z}\Big|}} \quad \text{ for $x, z \in \D$},
\end{equation}
and recall that the disk $\D$ endowed with the distance $d_\D(\cdot, \cdot)$ is the Poincar{\'e} model for the Lobachevsky plane.
\begin{proposition}\label{prop-intro-single-bergman}
Fix $f \in A^2(\D)$, $z \in \D$ and a  sequence $(s_n)_{n\ge 1}$ such that $s_n>  1$ and  $\sum_{n=1}^\infty (s_n-1)^2 <\infty$.
For $\Pi_{K_{\D}}$-almost any $X$ we have
\begin{align}\label{rec-single-f-intro}
 f(z) = \lim_{n\to\infty} \frac{\displaystyle{\sum_{k=0}^\infty \sum_{x \in X \atop k\le d_\D (z,x) < k+1} e^{-s_n d_\D(z, x)} f(x)}}{\displaystyle{  \sum_{x \in X} e^{-s_n d_\D(z,x)} }} .\end{align}
\end{proposition}
\begin{remark} We shall check that
for $\Pi_{K_{\D}}$-almost any $X$ and any $s_n>1$ we have
\[
\sum_{k=0}^\infty \Big| \sum_{x \in X \atop k \le d_\D(x, z) < k+1} e^{-s_n d_\D(z, x)} f(x)\Big|<\infty.
\]
The double summation in the numerator of the right hand side of  the equality \eqref{rec-single-f-intro} seems to be necessary since, for a general Bergman function $f\in A^2(\D)$, the absolute convergence of the series $\sum_{x \in X } e^{-s d_\D(x, z)} f(x)$ is not clear, cf. Remark \ref{no-abs-conv} below.
\end{remark}
Proposition \ref{prop-intro-single-bergman} will be derived from the more general Proposition \ref{prop-single-f} below.
For $\Pi_{K_\D}$-almost any $X\in \Conf(\D)$, simultaneous reconstruction of a  family of functions $f $ from their restrictions $f|_{X}$ will require more effort.
 The almost sure statement in Proposition \ref{prop-intro-single-bergman} is of course immediately extended to a countable dense family $\mathscr{F} \subset A^2(\D)$ of functions $f$ and a countable dense subset $\mathcal{A} \subset \D$ of points $z$. Therefore, by continuity of holomorphic functions, for $\Pi_{K_\D}$-almost every $X$, each function $f$ in this countable family $\mathscr{F}\subset A^2(\D)$ can be reconstructed from its restriction onto $X$. The implied subset of full measure  in our statement might however depend on the chosen countable family $\mathscr{F}$ and the limit equality \eqref{rec-single-f-intro} does not allow us to pass from $\mathscr{F}$ to  its closure $\overline{\mathscr{F}}^{A^2(\D)}    = A^2(\D)$. Nevertheless, we will show that extrapolation is possible for  the harmonic Hardy space $h^2(\D)$ whose definition we now recall:
\begin{align*}
h^2(\D) : &= \left\{f: \D\rightarrow \C\Big| \text{$f$ is harmonic and $ \| f\|_{h^2(\D)}^2 =  \sup_{0< r <1} \int_0^{2 \pi} |f(r e^{i \theta})|^2 \frac{d\theta}{2\pi} <\infty$}\right\}.
\end{align*}

\begin{theorem}\label{thm-disk} For
 $\Pi_{K_\D}$-almost every $X$,  we have for all $f \in h^2(\D)$, all $z \in \D$ and all $s> 1$, the series $
\sum_{x\in X} e^{-s d_\D(z,x)} f(x)$ converges absolutely and
\begin{align}\label{rec-hardy-no-subseq}
 f(z)    = \lim_{s \to 1^{+}} \frac{  \displaystyle{ \sum_{x \in X} e^{-s d_\D(z, x)} f(x)}}{\displaystyle{  \sum_{x \in X} e^{-s d_\D(z,x)} }}.
\end{align}
\end{theorem}

Theorem \ref{thm-disk} will be derived from the more general
Theorem \ref{thm-intro-no-subseq} below. As we shall see, even for a fixed $f\in h^2(\D)$, the passage from the limit equality \eqref{rec-single-f-intro} using a fixed sequence $s_n$ approaching $1^{+}$ to the limit equality \eqref{rec-hardy-no-subseq} using the limit $s \to 1^{+}$  requires some effort.   Theorem \ref{thm-disk} implies that $\Pi_{K_\D}$-almost every $X$ is a uniqueness set for $h^2(\D)$, but  does not imply the result of \cite{BQS-LP} that  $\Pi_{K_\D}$-almost every $X$ is a uniqueness set for $A^2(\D)$.
The impossibility of reconstruction of all Bergman functions simultaneously using averaging with radial weights is proved in \S \ref{sec-failure}. It would be interesting to extend our results to determinantal point processes corresponding to weighted Bergman spaces, cf. \cite{BQ-CMP}.

\subsubsection{The assumption on the variance of linear statistics}
Let $E$ be a complete separable metric  space, and let  $\Pi$ be a point process on $E$; precise definitions and background
on point processes are recalled in  \S \ref{sec-pp} of the Appendix.
 Our main assumption on $\Pi$ is the following upper bound \eqref{var-less-2-moment} on the variance of linear statistics.

\begin{assumption}[A bound on the variance]\label{ass-pp}There exists a constant $C$ depending only on $\Pi$ such that
for any function $f: E\rightarrow \C$ satisfying $\E_\Pi [\sum_{x\in \X} | f(x)| + | f(x)|^2] <\infty$, we have
\begin{align}\label{var-less-2-moment}
\Var_\Pi\Big(\sum_{x\in \X} f(x)\Big) \le C   \cdot \E_\Pi \Big( \sum_{x\in \X} | f(x)|^2\Big).
\end{align}
\end{assumption}

More generally, given a Hilbert space $\mathscr{H}$, Assumption \ref{ass-pp} immediately implies
\begin{proposition}\label{rem-scalar-vector}
The  inequality \eqref{var-less-2-moment}  holds, with the same constant,  for any Bochner measurable function $f: E \rightarrow \mathscr{H}$  such that $\E_\Pi [\sum_{x\in \X} \| f(x)\|_{\mathscr{H}} + \| f(x)\|_{\mathscr{H}}^2] <\infty$.
\end{proposition}

Fix a  reference Radon measure $\mu$   on $E$, assume that the correlation measures of $\Pi$ are absolutely continuous with respect to  corresponding tensor powers of $\mu$, let   $\rho_1^{(\Pi)}$ and $\rho_2^{(\Pi)}$ be the first and the second order correlation functions  of $\Pi$ with respect to  $\mu$.  We say that $\Pi$ is negatively correlated, if  $\rho_2^{(\Pi)}(x, y) \le \rho_1^{(\Pi)}(x) \rho_1^{(\Pi)}(y)$ for $(x, y) \in E\times E$.

\begin{lemma}\label{lem-var-LS}
A negatively correlated point process satisfies Assumption \ref{ass-pp}.
\end{lemma}

The routine proof of Lemma \ref{lem-var-LS} is recalled in the Appendix.  Lemma \ref{lem-var-LS} implies that Poisson point processes as well as determinantal point processes induced by Hermitian correlation kernels satisfy Assumption \ref{ass-pp}.

\subsubsection{The main result for the  disk}
Recall that the line element for the distance (\ref{lob-dist}) is
\[
ds^2 =  \frac{4| dz|^2}{(1 - |z|^2)^2},
\]
and  the corresponding volume measure $\mu_\D$ on $\D$ is
\[
d \mu_\D(z) = \frac{dV(z)}{(1 - |z|^2)^2}.
\]
Let $\T=\partial \D$.  Recall that the Poisson kernel  $P: \D \times \T \rightarrow \R^{+}$ is given by the formula
\[
P(z, \zeta)=   \frac{1- |z|^2}{ |1 - \bar{z} \zeta|^2} =  \frac{1- |z|^2}{ |z - \zeta|^2}.
\]
Write $L^2(\T)  = L^2(\T, \frac{d\theta}{2\pi})$. For $z\in \D$,  introduce a function $P_z:\T\to \mathbb R$,
$
P_z(\zeta) = P(z, \zeta).
$

Let $\mathfrak{B}$ be a Banach space, let $x_n \in \mathfrak{B}$. Recall that a convergent series  $\sum_{n=1}^\infty x_n$  is said to converge
absolutely if $\sum_{n=1}^\infty \| x_n\|_{\mathfrak{B}}<\infty$ and to converge unconditionally if its sum does not change under any reordering of the terms.
For example, under some additional assumptions, see Proposition \ref{prop-positive-uncond} below, a convergent functional series of positive  functions converges unconditionally.
\begin{theorem}\label{thm-intro-no-subseq}
If $\Pi$ is a point process on $\D$ satisfying  Assumption \ref{ass-pp} and having first intensity measure $\lambda \mu_\D$ with $\lambda> 0$ a constant, then
for $\Pi$-almost any $X \in \Conf(\D)$ we have:
\begin{enumerate}
\item For any $s>1$ and any $z\in \D$, the series $\sum_{x\in X} e^{-s d_\D(x, z)} P_x$ converges unconditionally in $L^2(\T)$.
\item  For any $z\in \D$, we have
\[
\lim_{s\to 1^{+}} \left\| \frac{\displaystyle{\sum_{x \in X}   e^{-s d_\D(x, z)} P_x}}{\displaystyle{  \sum_{x \in X}  e^{-s d_\D(x, z)} }} - P_{z} \right\|_{L^2(\T)} = 0.
\]
\item For any $ f  \in h^2(\D)$, any $z\in \D$ and any $s >1$, the series  $\sum_{x \in X}   e^{-s d_\D(x, z)} f(x)$ converges absolutely  and
\[
f(z) =    \lim_{s\to 1^{+}}  \frac{ \displaystyle{ \sum_{x \in X}   e^{-s d_\D(x, z)} f(x) } }{\displaystyle{  \sum_{x \in X}   e^{-s d_\D(x, z)} }}.
\]
\end{enumerate}
\end{theorem}

\begin{remark} \label{no-abs-conv}
If $\Pi$ is a Poisson point process with intensity measure $\mu_\D$ and $1 < s \le 3/2$, then the Kolmogorov Three Series Theorem implies that  the series $\sum_{x \in X} e^{-s d_\D(x, z)} \| P_x\|_{L^2(\T)}$ diverges for $\Pi$-almost every $X$ and all $z\in \D$.
\end{remark}

\subsection{Point processes on the complex hyperbolic space}
Let $d\in \mathbb N$.  For $z, w \in \C^d$, write $z \cdot w  = \sum_{k=1}^d z_k w_k;  \bar{z} = (\bar{z}_1, \cdots, \bar{z}_d);  |z| = \sqrt{ z \cdot \bar{z}}$.
Let $\D_d  = \{z  \in \C^d: |z|  < 1\}$ be the unit ball in $\C^d$ endowed with the Lebesgue measure $dV$.
Recall that any bounded complex domain carries a natural Riemannian metric, called the Bergman metric, cf. e.g. Krantz \cite[Chapter 1]{Krantz},  defined in terms of the reproducing kernel of the space of square-integrable holomorphic functions on our domain and thus, by definition, invariant under biholomorphisms. In the particular case of $\D_d$, the Bergman metric takes the form
\[
d s_B^2 : =  4 \frac{| dz_1|^2 + \cdots + |dz_d|^2}{1 - |z|^2} + 4 \frac{| z_1 dz_1 + \cdots  + z_d dz_d|^2 }{(1 - |z|^2)^2}.
\]
Let $d_B(\cdot, \cdot)$ denote the distance under the Bergman metric. The ball $\D_d$ endowed with the metric $d_B$ is a model for the complex hyperbolic space.
For $w \in \D_d \setminus \{0\}$, set
\begin{align}\label{inv-auto}
\varphi_w(z) := \frac{w - \frac{z \cdot \bar{w}}{|w|^2}w - \sqrt{1 - |w|^2}\big(z - \frac{z \cdot \bar{w}}{|w|^2} w\big)  }{1 - z \cdot \bar{w}}.
\end{align}
For $w = 0$, set $\varphi_w(z) = - z$.   By \cite[Theorem 2.2.2]{Rudin-ball}, the map $\varphi_w$ defines a biholomorphic involution of $\D_d$
interchanging $w$ and $0$: we have $\varphi_w(0) = w, \varphi_w(w) = 0, \varphi_w (\varphi_w (z)) = z$ for all $z\in \D_d$. For any $z, w \in \D_d$, we have
\begin{align}\label{Berg-metric-def}
d_B(z, w) = \log \left(\frac{1 +  |\varphi_w(z)|}{ 1- | \varphi_w(z)|}\right).
\end{align}
The volume measure $\mu_{\D_d}$  associated to the Bergman metric is
\begin{align}\label{Berg-vol-def}
d\mu_{\D_d} (z) = \frac{ d V(z)}{(1 - |z|^2)^{d+1}},
\end{align}
and the Bergman Laplacian  $\widetilde{\Delta}$ is given by the formula
\[
\widetilde{\Delta} = (1 - |z|^2) \sum_{i, j } (\delta_{ij} - z_i \bar{z}_j) \frac{\partial^2}{\partial z_i \partial\bar{z}_j}.
\]
Recall that a function $f \in C^2(\D_d)$ is called  $\MM$-harmonic if $\widetilde{\Delta} f\equiv 0$ on $\D_d$, and
denote by $\MM\HH(\D_d)$ the set of all complex-valued $\MM$-harmonic functions on $\D_d$.

Set $\Sph_d = \{z \in \C^d: |z|=1\}$, let  $\sigma_{\Sph_d}$ be the normalized surface measure on $\Sph_d$, write $L^2(\Sph_d) = L^2(\Sph_d, \sigma_{\Sph_d})$.
Let $\MM\HH^2(\D_d)$ be the $\MM$-harmonic Hardy space, cf. e.g. \cite{ABC-harmonic}, \cite{Bonami-hardy}:
\begin{equation}\label{def-M-hardy}
\MM\HH^2(\D_d): = \Big\{f \in \MM \HH(\D_d) \Big|  \| f\|_{\MM\HH^2(\D_d)}^2 =  \sup_{0< r <1} \int_{\partial \D_d}|f(r \zeta)|^2 d\sigma_{\Sph_d}(\zeta) <\infty \Big\}.
\end{equation}

 The Poisson-Szeg\"o kernel $P^b: \D_d \times \Sph_d \rightarrow \R^{+}$  is defined by the formula
\begin{align}\label{def-Poisson-Szego}
P^b(w, \zeta)  = \frac{(1-|w|^2)^d}{|1 - \zeta\cdot \bar{w}|^{2d}}, \quad w \in \D_d, \zeta \in \Sph_d.
\end{align}
For any fixed $ \zeta \in \Sph_d$, the function $w \mapsto P^b(w, \zeta)$ is $\MM$-harmonic on $\D_d$.
In what follows, for any $w \in \D_d$, we denote
\begin{equation}\label{def-pwb}
P_w^b(\zeta) = P^b(w, \zeta) \quad \zeta \in \Sph_d.
\end{equation}
For any function $f \in \MM\HH^2(\D_d)$, there exists, see e.g. \cite[Proposition 2.4]{ABC-harmonic},  a unique $g \in L^2(\Sph_d) = L^2(\Sph_d, \sigma_{\Sph_d})$ such that
\[
f (w) = P^b[g](w): = \int_{\Sph_d} P^b(w, \zeta) g(\zeta)  d\sigma_{\Sph_d}(\zeta).
\]

\begin{theorem}\label{thm-intro-hardy-ball}
If  a point process $\Pi$ on $\D_d$ satisfies  Assumption \ref{ass-pp} and has first intensity measure $\lambda \mu_{\D_d}$ with $\lambda>0$ a constant, then, for $\Pi$-almost every $X\in \Conf(\D_d)$, we have:
\begin{enumerate}
\item For any $s>d$ and any $z\in \D_d$, the series $\sum_{x\in X} e^{-s d_B(x, z)} P_x^b$ converges unconditionally in $L^2(\Sph_d)$.
\item  For any $z\in \D_d$, the functions (\ref{def-pwb}) satisfy
\[
\lim_{s\to d^{+}} \left\| \frac{\displaystyle{\sum_{x \in X}   e^{-s d_B(x, z)} P_x^b}}{\displaystyle{  \sum_{x \in X}  e^{-s d_B(x, z)} }} - P_{z}^b \right\|_{L^2(\Sph_d)} = 0.
\]
\item For any $f  \in \MM\HH^2(\D_d)$, any $z\in \D_d$ and  any $s > d$, the series  $\sum_{x \in X}   e^{-s d_B(x, z)} f(x)$ converges absolutely and satisfies
\[
f(z) =    \lim_{s\to d^{+}}  \frac{ \displaystyle{ \sum_{x \in X}   e^{-s d_B(x, z)} f(x) } }{\displaystyle{  \sum_{x \in X}   e^{-s d_B(x, z)} }}.
\]
\end{enumerate}
\end{theorem}
\subsection{Point processes on the real hyperbolic space}
Let $m  \ge 2$ be a positive integer and let $\B_m \subset \R^m$ be the open unit ball  endowed with the Lebesgue measure $dV$. The Poincar\'e metric on $\B_m$, see e.g. Stoll \cite{Stoll-hyper-ball}, is defined by the formula
\begin{equation}\label{metric-real-ball}
ds_h^2 =  4 \frac{dx_1^2 + \cdots + dx_m^2}{(1 - |x|^2)^2}.
\end{equation}
Let $d_h(\cdot, \cdot)$ denote the distance under the Poincar\'e metric.  The ball $\B_m$ endowed with the metric $d_h$ is a model for the $m$-dimensional Lobachevsky space.  For any $a\in \B_m$, set
\begin{align}\label{def-psi-a}
\psi_a (x): =  \frac{a | x-a|^2  + (1 - |a|^2) (a - x)}{ | x- a|^2 + ( 1- |a|^2) (1 - |x|^2)}.
\end{align}
By \cite[Theorem 2.1.2, Theorem 2.2.1]{Stoll-hyper-ball}, the map $\psi_a$ is an involutive isometry of $\B_m$ interchanging $0$ and $a$: we have
$$\psi_a(0) = a, \psi_a(a) = 0, \psi_a (\psi_a (x)) = x$$ for all $x\in \B_m$.  For $a,b\in \B_m$ we have
\begin{align}\label{def-d-h}
d_h(a, b) = \log \left(  \frac{1 + | \psi_a(b) | }{1  - | \psi_a(b)|}\right).
\end{align}
The volume measure $\mu_{\B_m}$ associated to the metric (\ref{metric-real-ball}) is
\begin{align}\label{hyper-vol-def}
d\mu_{\B_m} (x) : = \frac{dV(x)}{(1 - |x|^2)^m},
\end{align}
and  the hyperbolic Laplacian $\Delta_h$ on $\B_m$ is:
\[
\Delta_h = ( 1 -|x|^2)^2 \sum_{i = 1}^m \frac{\partial^2}{\partial x_i^2} + 2 (m-2) (1  - |x|^2) \sum_{i = 1}^m x_i \frac{\partial }{\partial x_i}.
\]
A function $f \in C^2(\B_m)$ satisfying $\Delta_h f \equiv 0$ is called $\HH$-harmonic. Let $\HH(\B_m)$ be the set of all complex-valued $\HH$-harmonic functions on $\B_m$.
Let $S^{m-1} = \partial \B_m$ be the unit sphere in $\R^m$, let $\sigma_{S^{m-1}}$ be the normalized surface measure on $S^{m-1}$,  write $$L^2(S^{m-1}) = L^2(S^{m-1}, \sigma_{S^{m-1}}).$$
Consider, cf. e.g. \cite{Stoll-hyper-ball}, the Hardy space
\begin{align}\label{def-intro-H-hardy}
\HH^2(\B_m): = \Big\{f \in \HH(\B_m)\Big|  \| f\|_{\HH^2(\B_m)}^2  = \sup_{0 < r < 1} \int_{S^{m-1}} | f(r t)|^2 d\sigma_{S^{m-1}} (t) <\infty\Big\}.
\end{align}
Recall that the hyperbolic Poisson kernel $P^h: \B_m \times S^{m-1} \rightarrow \R^{+}$  of the unit ball $\B_m$ is
\begin{align}\label{def-hyper-p-kernel}
P^h (x, t) = \left(\frac{ 1 - |x|^2}{| x - t|^2}\right)^{m-1}.
\end{align}
For any fixed $t \in S^{m-1}$, the function $x \mapsto P^h (x, t)$ is $\HH$-harmonic on $\B_m$.
For $x\in \B_m$, set
\begin{equation}\label{def-phx}
P^h_x(t) = P^h(x, t) \quad t \in S^{m-1}.
\end{equation}
For any $f \in \HH^2(\B_m)$, by \cite[Theorem 7.1.1]{Stoll-hyper-ball}, there is a unique $g \in L^2(S^{m-1})$ such that
\begin{align}\label{hyper-Poisson-transform-1}
f (x) = P^h[g](x): = \int_{S^{m-1}} P^h(x, t) g(t)  d\sigma_{S^{m-1}}(t).
\end{align}

\begin{theorem}\label{thm-intro-hyperball}
If  a point process $\Pi$ on $\B_m$ satisfies  Assumption \ref{ass-pp} and has first intensity measure $\lambda \mu_{\B_m}$ with $\lambda>0$ a constant, then, for $\Pi$-almost every $X\in \Conf(\B_m)$  we have:
\begin{enumerate}
\item For any $s>m-1$ and any $a\in \B_m$, the series $\sum_{x\in X} e^{-s d_h(x, a)} P_x^h$ converges unconditionally in $L^2(S^{m-1})$.
\item  For any $a \in \B_m$, the functions (\ref{def-phx}) satisfy
\[
\lim_{s\to (m-1)^{+}} \left\| \frac{\displaystyle{\sum_{x \in X}   e^{-s d_h(x, a)} P_x^h}}{\displaystyle{  \sum_{x \in X}  e^{-s d_h(x, a)} }} - P_{a}^h \right\|_{L^2(S^{m-1})} = 0.
\]
\item For any $ f  \in \HH^2(\B_m)$, any $ a\in \B_m$ and any $s > m-1$, the series  $\sum_{x \in X}   e^{-s d_h(x, a)} f(x)$ converges absolutely    and satisfies
\[
f(a) =    \lim_{s\to (m-1)^{+}}  \frac{ \displaystyle{ \sum_{x \in X}   e^{-s d_h(x, a)} f(x) } }{\displaystyle{  \sum_{x \in X}   e^{-s d_h(x, a)} }}.
\]
\end{enumerate}
\end{theorem}

\subsection{Patterson-Sullivan measures for point processes on $\catone$ spaces}
\subsubsection{The Busemann boundary}\label{sec-Buse-bd}
 Let $(M, d)$ be a proper  complete metric space. The group of isometries $\Iso(M) = \Iso(M, d)$ of $M$  is locally compact and separable with respect to the compact-open topology. We fix a base point $o\in M$; the definitions and the results below of course do not depend on the specific choice of the base point.
Let $C(M)$ be the space of continuous functions on $M$ equipped with the topology of uniform convergence on compact subsets. Let $C_{*}(M) = C(M)/\{\text{constant functions}\}$, equipped with the quotient topology, namely,  $[f_i] \to [f]$ in $C_{*}(M)$ if and only if there exist $a_i \in \C$ such that $f_i + a_i \to f$ in $C(M)$. Recall, cf.  e.g. \cite[\S 3.1]{Gromov-lecture}, that the horofunction compactification $\overline{M}$ of $M$  is the closure of the image of the  injective map
$
M  \longrightarrow  C_{*}(M),  \ x \mapsto  [d_x],
$
where $d_x$ is the distance function $d_x(\cdot) = d(x, \cdot)$.
Introduce the Busemann boundary $\partial M : = \overline{M}\setminus M$.  For  $\xi \in \partial M, x, y \in M$, the additive {\it Busemann cocycle}
$
B_\xi (x, y)$
 is the continuous extension to $\partial M \times M \times M$ of the function $ M \times M \times M \ni (z, x, y) \mapsto  B_z(x, y) = d(z, x) - d(z, y)$.
Recall that for a  complete $\catzero$ space,  the  Gromov boundary  and  the Busemann boundary coincide, see e.g. Bridson and Haefliger  \cite[Theorem 8.13]{Bridson-Haefliger}.
\subsubsection{Conformal density}
The action of $\Iso(M)$ on $M$ extends to a continuous action on $\overline{M}$ by homeomorphisms. Let $G< \Iso(M)$ be a closed subgroup. Denote by $\mathfrak{M}^{+}(\partial M)$ the set of positive Radon measures on $\partial M$ and by $\mathfrak{M}^{1}(\partial M)$ the subset of probability measures on $\partial M$.  A $G$-conformal density of dimension $\delta> 0$ is a  continuous map
$$\nu: M  \longrightarrow  \mathfrak{M}^{+}(\partial M),
x  \mapsto  \nu_x,
$$
$G$-equivariant in the sense that $g_{*}\nu_x   = \nu_{gx}$ for all $g \in G, x \in M$, and such that all measures $\nu_x$ are
mutually absolutely continuous with Radon-Nikodym derivatives  given by the identity
\[
\frac{d\nu_x}{d\nu_y}(\xi) = e^{- \delta B_\xi(x,  y)}  \quad \text{for all $x, y \in M$ and $\nu_x$-almost all $\xi\in \partial M$.}
\]

Set
\begin{equation}\label{def-nu-bar}
\bar{\nu}_x = \frac{\nu_x}{\nu_x(\partial M)}.
\end{equation}
Our family $(\nu_x)_{x\in M}$ is called normalized with respect to the  base point $o \in M$ if $\nu_o = \bar{\nu}_o$.

For $x\in M, r >0$, consider the open ball $B(x, r) = \{y\in M| d(y, x)<r\}$. We start with an auxiliary
\begin{proposition}\label{prop-exp-cv-intro}
If $M$ is a proper geodesic $\catone$ Riemannian manifold endowed with the  volume measure $\mu_M$ and admitting a discrete non-elementary subgroup $G< \Iso(M)$ acting cocompactly on $M$, then
\begin{enumerate}
\item there exist a constant $h_M >0$ and a positive constant $C\ge 1$ such that
\begin{align}\label{ass-rate-cat-1}
C^{-1} e^{r \cdot h_M } \le \mu_M(B(o, r)) \le   C e^{r\cdot h_M} \quad \text{for all $r >0$};
\end{align}
\item  there exists a probability measure $\mu_o$ on $\partial M$, such that the following weak convergence of probability measures on $\overline{M}$ holds:
\begin{align}\label{def-mu-o-cat-1}
\lim_{ s \to  h_M^{+}}  \frac{ e^{-s d(x, o)} \mu_M(dx)}{ \displaystyle \int_M e^{-s d(x, o)} d\mu_M(x)}  =  \mu_o;
\end{align}
\item  the unique $\Iso(M)$-conformal density of dimension $h_M$ is given by the map
\begin{equation}\label{def-conf-density}
\mu: M  \longrightarrow  \mathfrak{M}^{+}(\partial M),
x  \mapsto  \mu_x = e^{- h_M B_\xi (x, o)} \mu_o (d\xi).
\end{equation}
\end{enumerate}
\end{proposition}
Proposition follows from the more general Proposition \ref{prop-exp-cv} below.
The constant $h_M$ in  \eqref{ass-rate-cat-1} is of course the volume entropy of $M$ if $(M, d)$ is a Riemannian manifold and $\mu_M$ is the associated volume measure.

\subsubsection{Weak convergence to the conformal density.}

 Given a configuration $X$ on $M$, we define its  critical exponent by the formula
\[
\delta_{\PS}(X): = \inf\Big\{s> 0\Big|\sum_{x \in X} e^{- s \cdot d(x, y)} <\infty\Big\} \in [0, \infty].
\]
The  definition  of $\delta_\PS(X)$ does not depend on the specific choice of $x, y \in M$. We say that a configuration  $X$ on $M$ is of divergent type if
\[
\delta_{\PS}(X)<\infty \text{\, and \,} \sum_{x\in X} e^{- \delta_{\PS}(X) \cdot d(o, x)} = \infty.
\]

\begin{theorem}\label{thm-cat-1}
If $M$ is a proper geodesic $\catone$ Riemannian manifold endowed with the  volume measure $\mu_M$, admitting a discrete non-elementary subgroup $G< \Iso(M)$ acting cocompactly on $M$,  and  $\Pi$ a point process on $M$ satisfying Assumption \ref{ass-pp} and having first intensity measure $\lambda \mu_M$ with  $\lambda > 0$ a constant, then $\Pi$-almost every configuration $X$ on $M$ is of divergent type with critical exponent $\delta_\PS(X)$ equal to the volume entropy $h_M$ of $M$, and, furthermore,  for any fixed sequence $\mathfrak{s}=(s_n)_{n\in\N}$ such that
\[
\sum_{n=1}^\infty (s_n- h_M)^2 < \infty,
\]
there exists a subset $\Omega_{\mathfrak{s}} \subset \Conf(M)$ satisfying $\Pi(\Omega_{\mathfrak{s}}) = 1$  such that for any $X \in \Omega_{\mathfrak{s}}$ and  any $y \in M$, we have
the following weak convergence of probability measures:
\[
\lim_{n\to\infty}    \frac{ \displaystyle \sum_{x\in X} e^{-s_n d(y, x)}  \delta_x }{ \displaystyle \sum_{x\in X} e^{- s_n d(y, x)}} = \overline{\mu}_y.
\]
\end{theorem}

Recall that $\overline{\mu}_y(d \xi)$ is defined by  (\ref{def-nu-bar}), (\ref{def-conf-density}).
Theorem \ref{thm-cat-1} follows from the more general Theorem \ref{thm-general} below.

\subsection{Reconstruction of harmonic functions}
Let $(M, d)$ be a proper  complete metric space. We fix a base point $o\in M$. As in \S \ref{sec-Buse-bd}, we denote by $\overline{M}   = M \cup  \partial M$ the horofunction compactification of $M$.  Let $\mu_M$ be a positive Radon measure on $M$.  We need the following assumption on the growth of the volume of the balls in $M$. Recall
 that   a positive function $f$ on $\R^{+}$ (or on $\N$) is  sub-exponential if for any $\alpha >0$, we have $\lim_{r\to + \infty} f(r) \exp(-\alpha r) = 0$.
A pair $(L, U)$, $L \le U$, of continuous  positive functions on $\R^{+}$ is called {\it controllable} if
$\liminf_{r\to \infty} L(r)> 0$  and the functions $L, U, \sup_{k \in \N} \frac{U(kr)}{L((k+1)r)}$ are all sub-exponential .
For example, if $0 < c< C< \infty$ and $\alpha \ge 0, \beta \ge 0$ and   $0 < \gamma < 1$, then the  pair  $(L, U)$ defined by $L(r) = c r^\alpha  \exp (\beta r^\gamma),  U(r) = C r^\alpha \exp (\beta r^\gamma)$, is  controllable.
For any $x\in M, r >0$, set $B(x, r) = \{y\in M| d(y, x)<r\}$. \begin{assumption}\label{ass-M-A1}
There exist $h_M >0$ and a controllable pair $(L, U)$  such that
\begin{align}\label{ass-rate}
L(r) e^{r \cdot h_M } \le \mu_M(B(o, r)) \le   U(r) e^{r\cdot h_M} \quad \text{for all $r >0$}.
\end{align}
\end{assumption}
In what follows, a key r{\^o}le is played by
\begin{assumption}\label{ass-M-A2}
There exists a probability measure $\mu_o$ on $\partial M$, such that the following weak convergence of probability measures on $\overline{M}$ holds:
\begin{align}\label{def-S-o}
\lim_{ s \to  h_M^{+}}  \frac{ e^{-s d(x, o)} \mu_M(dx)}{ \displaystyle \int_M e^{-s d(x, o)} d\mu_M(x)}  =  \mu_o (d\xi).
\end{align}
\end{assumption}
A  simple particular case is given by
\begin{proposition}\label{prop-unique-rot-inv}
Let  $\mu_M$ be a positive Radon measure satisfying Assumption \ref{ass-M-A1}. Let $G_o<\Iso(M)$ be the subgroup preserving the measure $\mu_M$ and
the point $o \in M$.  If $G_o$ acts transitively on $\partial M$, then   \eqref{def-S-o} holds, and the measure $\mu_o$   on $ \partial M$ is obtained by choosing an arbitrary $\xi\in\partial M$ and taking the image of the normalized Haar measure on $G_o$ under the map $g \mapsto g (\xi) $.
\end{proposition}
 Define a positive kernel $\mathcal{P}: M \times \partial M \rightarrow \R^{+}$ by
\begin{align}\label{def-abstract-P-kernel}
\mathcal{P}(x, \xi): = \frac{\displaystyle   e^{- h_M B_\xi (x, o)}}{\displaystyle  \int_{\partial M} e^{- h_M B_\xi (x, o)} d\mu_o (\xi)}.
\end{align}
For $x\in M$, introduce a function $\mathcal{P}_x$ on $\partial M$ by the formula
\begin{equation}\label{p-x-xi}\mathcal{P}_x(\xi) = \mathcal{P}(x, \xi).
\end{equation}
Set $L^2(\partial M) = L^2(\partial M, \mu_o)$.
For $g \in L^2(\partial M)$,  $x\in M$, set
\[
\mathcal{P}[g](x) := \int_{\partial M} \mathcal{P}(x, \xi) g(\xi) d\mu_o(\xi) \quad x\in M;
\]
\begin{align}\label{def-abstract-hardy}
\HH^2_\mathcal{P} (M): = \Big\{f = \mathcal{P}[g]\Big| g \in L^2(\partial M) \Big\}.
\end{align}

\begin{assumption}[Mean value property]\label{ass-MVP}
For any $\xi \in \partial M$ and any  $z \in M$, we have
\[
\mathcal{P}(z, \xi) = \frac{1}{\mu_M(B(z, r))}\int_{B(z, r)} \mathcal{P}(x, \xi) d\mu_M(x) \quad \text{for all $r> 0$.}
\]
\end{assumption}

\begin{assumption}[$L^2$-growth rate of the kernel]\label{ass-P-growth}
There exists a non-decreasing sub-exponential function $\Theta : \R^{+}\rightarrow \R^{+}$ such that
\begin{align}\label{2-norm-poi-sze}
\| \mathcal{P}_x\|_{L^2(\partial M)}^2  = \int_{\partial M} \mathcal{P} (x, \xi)^2 d\mu_o(\xi) \le  \Theta( d(o, x))\cdot e^{h_M d(o, x)}.
\end{align}
\end{assumption}

\begin{theorem}\label{thm-rec-M}
Consider a triple  $(M, d, \mu_M)$ satisfying Assumptions \ref{ass-M-A1}, \ref{ass-M-A2}, \ref{ass-MVP} and \ref{ass-P-growth}. If a point process $\Pi$ on $M$ with first intensity measure $\mu_M$ satisfies Assumption \ref{ass-pp}, and a sequence $(s_n)_{n\ge 1}$ in $(h_M, \infty)$ satisfies $\sum_{n=1}^\infty (s_n - h_M) <\infty$, then for $\Pi$-almost every $X$ we have:
\begin{enumerate}
\item For any $s>h_M$ and any $z\in M$, the series $\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x$ converges unconditionally in $L^2(\partial M)$.
\item  For any $z\in M$, the function \eqref{p-x-xi} satisfies
\begin{align}\label{thm-rec-M-goal}
\lim_{n\to\infty} \left\| \frac{\displaystyle{\sum_{x \in X}   e^{-s_n d(x, z)} \mathcal{P}_x}}{\displaystyle{  \sum_{x \in X}  e^{-s_n d(x, z)} }} - \mathcal{P}_{z} \right\|_{L^2(\partial M)} = 0.
\end{align}
\end{enumerate}
\end{theorem}
Since, for a scalar series,  unconditional convergence implies absolute convergence,  it follows from Theorem \ref{thm-rec-M}  that for $\Pi$-almost every $X$, any $ f  \in \HH_\mathcal{P}^2(M)$, any $z\in M$ and any $n\in\N$, the series  $\sum_{x \in X}   e^{-s_n d(x, z)} f(x)$ converges absolutely, and
\[
f(z) =    \lim_{n\to\infty}  \frac{ \displaystyle{ \sum_{x \in X}   e^{-s_n d(x, z)} f(x) } }{\displaystyle{  \sum_{x \in X}   e^{-s_n d(x, z)} }}.
\]
Assuming an asymptotic for the volume of a ball, we obtain a stronger result.
\begin{assumption}\label{ass-M-pvg}
There exist constants $h_M>0$,  $\beta \ge 0$ and, for any $z\in M$, $c_z>0$, such that
\[
\lim_{r\to\infty} \frac{\mu_M(B(z, r))}{c_z r^\beta e^{rh_M}} = 1.
\]
\end{assumption}
\begin{theorem}\label{thm-rec-M-bis}
Consider a triple  $(M, d, \mu_M)$  satisfying Assumptions \ref{ass-M-A2}, \ref{ass-MVP},  \ref{ass-P-growth} and \ref{ass-M-pvg}. If a point process $\Pi$ on $M$ with first intensity measure $\mu_M$ satisfies Assumption \ref{ass-pp}, then for $\Pi$-almost every $X$ we have:
\begin{enumerate}
\item For any $s>h_M$ and any $z\in M$, the series $\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x$ converges unconditionally in $L^2(\partial M)$.
\item  For any $z\in M$, the function \eqref{p-x-xi} satisfies
\begin{align}\label{thm-rec-M-bis-goal}
\lim_{s\to h_M^{+}} \left\| \frac{\displaystyle{\sum_{x \in X}   e^{-s d(x, z)} \mathcal{P}_x}}{\displaystyle{  \sum_{x \in X}  e^{-s d(x, z)} }} - \mathcal{P}_{z} \right\|_{L^2(\partial M)} = 0.
\end{align}
\end{enumerate}
\end{theorem}
As above, it follows that for $\Pi$-almost all $X$ we have for any $ f  \in \HH_{\mathcal{P}}^2(M)$, any $z\in M$ and any $s >h_M$, the series  $\sum_{x \in X}   e^{-s d(x, z)} f(x)$
converges absolutely and
\[
f(z) =    \lim_{s\to h_M^{+}}  \frac{ \displaystyle{ \sum_{x \in X}   e^{-s d(x, z)} f(x) } }{\displaystyle{  \sum_{x \in X}   e^{-s d(x, z)} }}.
\]

\begin{remark}
In this paper we limit ourselves to the real and the complex hyperbolic spaces, but we expect the above formalism  to apply more generally, in particular, to more general symmetric spaces.
\end{remark}

\subsection{An outline of the argument}\label{sec-disk}

 For concreteness, here we consider the disk; the case of higher dimension is similar.
Consider a Banach space  $\mathfrak{F}$ of harmonic functions on $\D$ and a  point process $\Pi$ on $\D$ having
first intensity measure $\lambda \mu_\D$ with $\lambda> 0$ a constant and such that $\Pi$-almost every configuration $X$ is
a set of uniqueness for $\mathfrak{F}$. We aim to find an explicit algorithm,  for $\Pi$-almost every configuration $X$,  of computing $f(z)$ from the restriction  $f|_{X}$, for {\it all} $f \in \mathfrak{F}$ and {\it all } $z \in \D$. In other words, we aim to  find an algorithm ${\rm ALG}(z, f, X)$ such that for $\Pi$-almost every configuration $X$, we have
\[
f(z) = {\rm ALG}(z, f, X), \quad \text{for all $f \in\mathfrak{F}$ and all $z\in \D$.}
\]
Our argument proceeds in three steps: we progressively show that for $\Pi$-almost every configuration, our algorithm works
\begin{itemize}
\item [(i)]   for a fixed $f \in \mathfrak{F}$ and a fixed $z\in \D$;
\item[(ii)] for all $f \in \mathfrak{F}$ and a fixed $z\in \D$;
\item[(iii)]  for all $f \in \mathfrak{F}$ and all $z\in \D$.
\end{itemize}
Proposition \ref{prop-single-f} below gives (i)  for the Bergman  space $A^2(\D)$.
The key point is the passage from the step (i) to the step (ii). Then one can pass from (ii) to  (iii)  by a limit transition as follows (although the limit transition presented does not allow us to obtain an explicit reconstruction formula as in Theorem \ref{thm-disk}):
if for any fixed $z\in \D$, there exists an algorithm ${\rm ALG}(z, \cdot, \cdot)$ such that for $\Pi$-almost every $X$ we have
\[
f(z) = {\rm ALG}(z, f, X),  \text{\,for all $f\in \mathfrak{F}$,}
\]
then, for any countable dense subset $\mathcal{A} \subset \D$, for $\Pi$-almost every $X$ we have
\[
f(z) = {\rm ALG}(z, f, X), \text{\,for all $f \in \mathfrak{F}$ and all $ z \in \mathcal{A}$.}
\]
Now take  $z_n\in \mathcal{A}$, $z_n \to z$, and,
for $\Pi$-almost every $X$ obtain
\[
f(z) = \lim_{n\to\infty} {\rm ALG}(z_n, f, X), \text{\,for all $f\in \mathfrak{F}$ and all $z\in \D$.}
\]
The reconstruction proceeds by taking weighted averages.
For us, a weight on $\D$ is  a locally integrable non-negative function $W: \D\rightarrow \R^{+}$, such that $\int_\D W(x) dV(x) \in (0, +\infty]$.
The weight $W$ is  called {\it radial} if $W(x) = W(|x|)$ for any $x\in \D$.
Take an index set $\Lambda \subset (\lambda_0, \infty)$ accumulating at $\lambda_0$.  Recall, cf. \eqref{inv-auto}, that $\varphi_z$, in the particular case of the disk, is the M{\"o}bius involution interchanging $0$ and $z$.
We aim to find a family of { radial} weights $(W_\lambda: \D \rightarrow \R^{+})_{\lambda \in \Lambda}$  such that for fixed $f\in \mathfrak{F}$ and fixed $z\in \D$, we have
 \begin{align}\label{analyse-rec}
\lim_{\Lambda \ni \lambda \to \lambda_0}   \underbrace{\frac{\displaystyle \sum_{x\in X} W_\lambda(\varphi_{z}(x)) f(x) }{\displaystyle \E_\Pi\left( \sum_{x\in \X} W_\lambda(\varphi_{z}(x))\right)} }_{\text{denoted by  $\mathscr{R}(W_\lambda, z,  f; X)$}} = f(z) \quad \text{for $\Pi$-almost every $X$.}
\end{align}
For  a harmonic function $f$ and a radial weight $W$ on $\D$ satisfying, for any $z\in \D$, the relations $W(\varphi_{z}(x)) \in L^1( \D,  d\mu_\D(x))$, $W(\varphi_{z}(x)) f(x) \in L^1( \D,  d\mu_\D(x))$, the mean-value property, cf. Lemma \ref{lem-observation} in the Appendix, gives
\begin{align}\label{eq-ratios}
f(z) = \E_\Pi (\mathscr{R}(W, z,  f; \X)) =  \frac{\displaystyle{\int_\D} W(\varphi_{z}(x)) f(x) d\mu_\D(x)}{\displaystyle{\int_\D} W(\varphi_{z}(x)) d \mu_\D(x)}.
\end{align}

For step (ii), let $\mathfrak{F}$ be a Banach space with the following reproducing type property:  there exists a Hilbert space $\widehat{\mathfrak{F}}$ and a harmonic map $\D \ni x \mapsto   P_\mathfrak{F}^x \in \widehat{\mathfrak{F}}$ such that any function $f \in \mathfrak{F}$ is of the form $f(x) = \langle g, P_\mathfrak{F}^x \rangle_{\widehat{\mathfrak{F}}}$  for all $x\in \D$,  where $g \in \widehat{\mathfrak{F}}$ and $\| g\|_{\widehat{\mathfrak{F}}} \le \| f \|_{\mathfrak{F}}$.
 The mean value property of harmonic functions gives $\E_\Pi \left( \mathscr{R}(W_\lambda, z,  P_\mathfrak{F}^{(\cdot)}; X) \right)  =  P_{\mathfrak{F}}^{z}$, whence
\begin{multline*}
\E_\Pi \left(  \sup_{f\in \mathfrak{F}: \| f\|_{\mathfrak{F}} \le 1 } \Big|  \mathscr{R}(W_\lambda, z,  f; \mathscr{Y})  -  \E_\Pi (\mathscr{R}(W_\lambda, z,  f; \X)) \Big|^2 \right)
\\
 \le \E_\Pi \left(   \left\|  \mathscr{R}(W_\lambda, z,  P_\mathfrak{F}^{(\cdot)}; \X) -  P_{\mathfrak{F}}^{z}  \right\|_{\widehat{\mathfrak{F}}}^2 \right)   = \Var_\Pi (\mathscr{R}(W_\lambda, z,  P_\mathfrak{F}^{(\cdot)}; \X) ),
\end{multline*}
and we make step (ii)  by proving
$$
 \lim_{\Lambda \ni \lambda \to \lambda_0} \Var_\Pi (\mathscr{R}(W_\lambda, z,  P_\mathfrak{F}^{(\cdot)}; \X) ) = 0.
$$
If $\mathfrak F$ is the space of uniformly continuous harmonic functions on $\D$, then one can of course pass from (i) to (ii) by a limit transition from a countable dense subset using  the inequality
\[
\left|  \mathscr{R}(W_\lambda, z,  f; X)  -  \mathscr{R}(W_\lambda, z,  \widetilde{f}; X)  \right| \le  \| f - \widetilde{f}\|_{\infty}  \cdot  \mathscr{R}(W_\lambda, z,  1; X);
\]
indeed, in this case, the norm of the evaluation functional at every point of the disk  is $1$. If, on the other hand,  $\mathfrak F$ is the Hardy space $h^2(\D)$, then $P_{\mathfrak{F}}^x \in L^2(\T)$ is defined through the Poisson-Szeg\"o kernel, and we have
\begin{equation}\label{lim-hardy-inf}
\lim_{|x| \to 1^{-}}||P_{\mathfrak{F}}^x||_{L^2(\T)}= \infty.
\end{equation}
The growth rate in \eqref{lim-hardy-inf} plays a key r\^ole in our argument, see Proposition \ref{prop-var-up-bd-M} below. For step (iii),  furthrmore, the exponential form of the weight is substantially used.

\subsection{Organization of the paper}
The paper is organized as follows. In \S\ref{sec-general-rec} we prove our general reconstruction results, Theorems \ref{thm-rec-M} and \ref{thm-rec-M-bis}.
In \S \ref{sec-comp-hyper}  from Theorems \ref{thm-rec-M}, \ref{thm-rec-M-bis}
we derive Theorem \ref{thm-intro-hardy-ball} on the reconstruction of Hardy functions on the complex hyperbolic space;  in \S \ref{sec-real-hyper},  a similar Theorem \ref{thm-intro-hyperball}  for the real hyperbolic space.  In \S \ref{sec-single-M} we extrapolate  a fixed  $\MM$-harmonic Bergman function, and, in  \S \ref{sec-single-H}, a fixed $\HH$-harmonic Bergman function.
In \S \ref{sec-more-disk}, we return to the determinantal point process $\Pi_{K_\D}$ induced by the Bergman kernel $K_{\D}$ on the disk.  Proposition \ref{thm-reconstruct-hardy} is devoted to the reconstruction of Hardy functions using  uniform ball averages. In Proposition \ref{prop-failure} we show that averaging with  compactly supported radial weights cannot reconstruct all  $A^2(\D)$-functions simultaneously; the proof relies on new formulas given in   Proposition \ref{prop-up-var-weight}, Proposition \ref{prop-new-var-f}  for the variance of vector-valued linear statistics under  $\Pi_{K_\D}$.  Theorem \ref{thm-cat-1} giving the convergence of Patterson-Sullivan measures  for point processes  on CAT(-1) spaces and a more general
Theorem \ref{thm-general} are proved in \S \ref{sec-PS}.  The appendix \S \ref{sec-appendix} contains  proofs of auxiliary  statements.

\noindent {\bf{Acknowledgements.}} Mikhael Gromov taught the Patterson-Sullivan theory to the older of us  in 1999; we are greatly indebted to him. We are deeply grateful to Alexander Borichev, S\'ebastien Gou\"ezel,  Pascal Hubert, Alexey Klimenko and Andrea Sambusetti for useful discussions. Part of this work was done during a visit to the Centro  De Giorgi della Scuola Normale Superiore di Pisa. We are deeply grateful to the Centre for its warm hospitality.
AB's research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 647133 (ICHAOS). YQ's research is supported by  the National Natural Science Foundation of China, grants NSFC Y7116335K1 and NSFC 11688101.

\section{Proof of Theorems \ref{thm-rec-M}, \ref{thm-rec-M-bis}.}\label{sec-general-rec}

\subsection{Proof of Theorem \ref{thm-rec-M}}\label{sec-proof-thm-rec-M}
Statement (1) of Theorem \ref{thm-rec-M} is proved in Proposition \ref{prop-uncond-cv}, statement (2) of Theorem \ref{thm-rec-M}   in Proposition \ref{prop-rec-M-bis}.
\subsubsection{Step 1}
Fix $s > h_M$ and $z\in M$. In Proposition \ref{prop-L2-sum-M} and Corollary \ref{cor-L1-sum-M} below we show that for $\Pi$-almost every $X$,  the
 series $\sum_{k=0}^\infty \sum_{x \in X  \atop k \le d(x, z) < k+1}   e^{-s d(x, z)} \mathcal{P}_x$ converges  absolutely in $L^2(\partial M)$. Write
\begin{align}\label{def-pv-M}
\sideset{}{^{v.p.}}\sum_{x\in X}  e^{-s d(x, z)}  \mathcal{P}_x: =  \sum_{k=0}^\infty \sum_{x \in X  \atop k \le d(x, z) < k+1}   e^{-s d(x, z)} \mathcal{P}_x=  \lim_{k \to\infty}  \sum_{x \in X, d(x, z) < k}   e^{-s d(x, z)} \mathcal{P}_x,
\end{align}
 the convergence taking place in $L^2(\partial M)$.
 For any configuration $X\in \Conf(M)$, any $s>0$ and $z \in M$,  let $T_k(s, z; X)$ denote the truncated $L^2(\partial M)$-valued linear statistics:
\begin{align}\label{trun-LS}
T_k(s, z; X): = \sum_{x \in X  \atop k \le d(x, z) < k+1}   e^{-s d(x, z)} \mathcal{P}_x.
\end{align}
For a Banach space $\mathfrak{B}$, let $L^2(\Conf(M), \Pi; \mathfrak{B})=L^2(\Pi; \mathfrak{B})$ be the Banach space of $\mathfrak{B}$-valued  functions defined on $\Conf(M)$ and square-integrable with respect to $\Pi$. We make a distinction between a fixed configuration and the configurations used as the integration variable by writing $X$ as a fixed configuration  and $\X$ as integration variable. For instance,  we will frequently use the notation
\[
\frac{\displaystyle \sum_{x \in X} f(x) }{\displaystyle \E_\Pi \Big( \sum_{x \in \X} f(x)  \Big)  }: = \frac{\displaystyle \sum_{x \in X} f(x) }{\displaystyle \int_{\Conf(E)} \sum_{x \in \X} f(x)   d \Pi (\X) }.
\]
\begin{proposition}\label{prop-L2-sum-M}
For $s> h_M$ and $z\in M$, we have
\begin{align}\label{prop-L2-sum-M-goal}
\sum_{k=0}^\infty    \Big\| T_k(s, z; \X)\Big\|_{L^2(\Pi; L^2(\partial M))} <\infty.
\end{align}
\end{proposition}

For $z\in M$, set
\begin{align}\label{def-omega-z}
\Omega_M(z): = \bigcap_{n\ge 1} \left\{  X\in \Conf(M): \sum_{k=0}^\infty  \Big\| T_k\Big(h_M + n^{-1}, z; X\Big) \Big\|_{L^2(\partial M)} <\infty \right\}.
\end{align}
Since the first moment of a random variable is dominated by  the second moment, Proposition \ref{prop-L2-sum-M} implies
\begin{corollary}\label{cor-L1-sum-M}
For $s> h_M$ and $z\in M$, we have
\begin{align}\label{cor-L1-sum-M-goal}
\sum_{k=0}^\infty    \E_\Pi \left(  \Big\|  T_k(s, z; \X)\Big\|_{L^2(\partial M)}  \right)<\infty
\end{align}
and  therefore
$
\Pi(\Omega_M(z)) = 1.
$
\end{corollary}

\begin{lemma}\label{lem-ball-av-M}
For any $s> 0, R>0$ and $z\in M$, we have
\[
\E_\Pi \left( \sum_{x\in \X \cap B(z, R)}  e^{-s d(z, x)}  \mathcal{P}_x \right) =  \mathcal{P}_z\cdot \E_\Pi \left( \sum_{x\in \X\cap B(z, R)}  e^{-s d(z, x)} \right).
\]
\end{lemma}
\begin{proof}
Using the identity
\[
e^{-s t} \mathds{1} ( t < R) = \int_0^R s e^{-s r} \mathds{1}(t< r) dr + \int_R^\infty s e^{-s r} \mathds{1}(t < R) dr
\]
and Assumption \ref{ass-MVP},  we obtain
\begin{multline*}
\E_\Pi \left( \sum_{x\in \X \cap B(z,R)}  e^{-s d(z, x)}  \mathcal{P}_x \right)  = \int_{M} e^{-s d(z, x)} \mathds{1}( d(z, x) < R)  \mathcal{P}_x d\mu_M(x)
\\
 = \int_M   \mathcal{P}_x d\mu_M(x) \int_0^R s e^{-s r} \mathds{1}( d(z, x) <r)  dr    + \int_M  \mathcal{P}_x d\mu_M(x) \int_R^\infty s e^{-s r} \mathds{1} (d(z, x) < R)  dr
\\
 = \int_0^R s e^{-s r} dr \int_{B(z, r)}  \mathcal{P}_x  d\mu_M(x)    +  \int_R^\infty s e^{-s r}  dr \int_{B(z, R)} \mathcal{P}_x d\mu_M(x)
\\
 = \mathcal{P}_z \cdot \left(  \int_0^R s e^{-s r} \mu_M(B(z, r))  dr +\int_R^\infty s e^{-sr}    \mu_M( B(z, R)) dr\right).
\end{multline*}
Similarly, we have
$$
\E_\Pi \left( \sum_{x\in \X \cap B(z,R)}  e^{-s d(z, x)}  \right)  = \int_0^R s e^{-s r} \mu_M(B(z, r))  dr +\int_R^\infty s e^{-sr}    \mu_M( B(z, R))dr.
$$
Combining the above two equalities, we complete the proof of the lemma.
\end{proof}
Lemma \ref{lem-ball-av-M} immediately implies
\begin{corollary}\label{cor-MVP}
For any $s> 0$ and $k\in \N$, we have
\[
\E_\Pi \left(   T_k(s, z; \X)\right) =  \mathcal{P}_z\cdot \E_\Pi \left( \sum_{x\in \X\atop k \le d(z, x) < k+1 }  e^{-s d(z, x)} \right).
\]
\end{corollary}
For $r>0, z \in M$, we have $B(z, r) \subset B(o, r + d(o, z))$, and Assumption \ref{ass-M-A1}  implies  that
for any $z\in M$ there exists  a sub-exponential function $U_z: \R^{+} \rightarrow \R^{+}$ such that
\begin{equation}\label{lem-ass-M-A3}
\mu_M(B(z, r)) \le U_z (r) e^{rh_M } \quad \text{for all $r>0$.}
\end{equation}

\begin{lemma}\label{lem-finite-exp-sum}
For any $s> h_M$  and any $z \in M$, we have $\E_{\Pi}[\sum_{x \in \X}  e^{-s d(z, x)}] <\infty$.
\end{lemma}
\begin{proof}
We have
\begin{multline*}
\E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(z, x)}\Big]   \le   e^{s d(o, z)}  \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big] \le e^{s d(o, z)}   \int_M e^{-s d(o, x)} d\mu_M(x)
\\
\le  e^{s d(o, z)}  \sum_{k = 0}^\infty e^{-s k} \mu_M( B(o, k+1)) \le e^{s d(o, z)} \sum_{k=0}^\infty e^{-s k } e^{h_M (k+1)} U(k+1),
\end{multline*}
and, since $U$ is sub-exponential, we have
$
\sum_{k=0}^\infty e^{- (s - h_M) k} U(k+1)<\infty.
$
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-L2-sum-M}]
Fix $s> h_M$, $z\in M$, set $A_k(z) : = \{x\in M| k \le d(x, z) < k+1  \}$. The truncated sum $T_k(s, z; X)$ defined in \eqref{trun-LS} satisfies
\begin{align*}
  \Big\|  T_k(s, z; \X)  \Big\|_{L^2(\Pi; L^2(\partial M))}^2     =      \Var_\Pi \left(  T_k(s, z; \X) \right)  + \left\|  \E_\Pi \left(  T_k(s, z; \X)\right) \right\|_{L^2(\partial M)}^2.
\end{align*}
Proposition \ref{rem-scalar-vector}, \eqref{2-norm-poi-sze} and \eqref{lem-ass-M-A3}  imply that, under Assumption \ref{ass-pp}, we have
\begin{multline*}
 \Var_\Pi \left(  T_k(s, z; \X)\right)
\le  C \int_{A_k(z)}  e^{-2s d(x, z)} \| \mathcal{P}_{x}\|_{L^2(\partial M)}^2 d\mu_M(x)
\\
 \le  C \int_{A_k(z)}  e^{-2s d(x, z)}    \Theta( d(x, o)) e^{h_M d(x, z)} d\mu_M(x)
\\
\le  C e^{h_M d(o, z)} \int_{A_k(z)} e^{- (2s -h_M) d(x, z)} \Theta(d(x, z) + d(z, o) ) d\mu_M (x)
\\
\le  C e^{h_M d(o, z)} e^{- (2s - h_M)k}  \Theta(k+1 + d(z, o)) \mu_M(A_k(z))
\\
\le  C e^{h_M d(o, z)} e^{- (2s - h_M)k}  \Theta(k+1 + d(z, o)) \mu_M(B(z, k+1)).
\end{multline*}
Consequently,  there exists $C_z>0$, such that
\[
 \Var_\Pi \left(   T_k(s, z; \X) \right)\le C_z e^{- (2s - 2h_M) k}   \Theta(k+1 + d(z, o)) U_z(k+1).
\]
Applying Corollary \ref{cor-MVP}, we obtain
 \begin{multline*}
\Big\|   T_k(s, z; \X)\Big\|_{L^2(\Pi; L^2(\partial M))} \le  \sqrt{C_z} e^{- (s - h_M)k } \sqrt{ \Theta(k+1 + d(z, o))\cdot  U_z(k+1)}
\\
 +   \| \mathcal{P}_z\|_{L^2(\partial M)}  \E_\Pi \left(  T_k(s, z; \X)\right).
\end{multline*}
Since $\Theta, U_z$ are sub-exponential, we have the convergence
\[
\sum_{k=0}^\infty e^{- (s - h_M)k } \sqrt{ \Theta(k+1 + d(z, o))\cdot U_z(k+1)} <\infty,
\]
which,  combined with Lemma \ref{lem-finite-exp-sum}, implies the desired convergence \eqref{prop-L2-sum-M}.
\end{proof}

\subsubsection{Step 2}
Fix $z\in M$ and consider the set  $\Omega_M(z)$ of \eqref{def-omega-z}.
\begin{proposition}\label{prop-uncond-cv}
For any $z\in M$, any $X\in \Omega_M(z)$, any $s>h_M$ and any $w\in M$, the series $\sum_{x\in X} e^{-s d(x,w)} \mathcal{P}_x$ converges unconditionally in $L^2(\partial M)$.
\end{proposition}

Our argument proceeds by establishing that
\begin{enumerate}
\item For any $s>h_M$, the series  \eqref{def-pv-M} converges absolutely in $L^2(\partial M)$.
\item For any $s>h_M$, the series $\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x$ converges unconditionally in $L^2(\partial M)$ and in particular we have the identity in $L^2(\partial M)$:
\begin{align}\label{eq-two-series}
\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x = \sideset{}{^{v.p.}}\sum_{x\in X}  e^{-s d(x, z)}  \mathcal{P}_x.
\end{align}
\end{enumerate}

It is not clear to us whether for any $X\in \Omega_M(z)$ and any $z'\in M, s>h_M$, the series $\sum_{k=0}^\infty \sum_{x\in X, k \le d(x, z') < k+1} e^{-s d(x, z')} \mathcal{P}_x$ converges absolutely in $L^2(\partial M)$ and whether there exists a $\Pi$-full measure subset $\widetilde{\Omega} \subset \Conf(M)$ such that for all $X\in \widetilde{\Omega}$, all $s> h_M$ and all $z \in M$, the series
\[
 \sideset{}{^{v.p.}}\sum_{x\in X}e^{-s d(x, z)} \mathcal{P}_x
\]
is well-defined. Nonetheless, the equality \eqref{eq-two-series} for fixed $z\in M$ and all $X$  in a  subset $\Omega_M(z)$ depending on $z$ and having full measure under $\Pi$ is sufficient for our purposes.

\begin{proof}[Proof of Proposition \ref{prop-uncond-cv}]
Fix $z\in M$ and  $X\in \Omega_M(z)$. From the positivity and the monotonicity in $s$  of  the summands $e^{-s d(x, z)} \mathcal{P}_x(\xi)$ it follows that if the series \eqref{def-pv-M} converges absolutely  in $L^2(\partial M)$ for an exponent $s>h_M$, then  so it does for all $s' \ge  s> h_M$. By  definition \eqref{def-omega-z} of $\Omega_M(z)$, the series \eqref{def-pv-M} converges absolutely in $L^2(\partial M)$ for all $s >h_M$.
For any $\xi \in \partial M$, we have $e^{-s d(x, z)} \mathcal{P}_x(\xi) \ge 0$, whence
\begin{align}\label{two-series-ptwise}
\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x(\xi) =  \sum_{k=0}^\infty \sum_{x \in X  \atop k \le d(x, z) < k+1}   e^{-s d(x, z)} \mathcal{P}_x(\xi) \in [0, \infty].
\end{align}
The right hand side of \eqref{two-series-ptwise} defines a function in $L^2(\partial M)$, consequently, so does the left hand side. The Dominated Convergence Theorem now implies the unconditional convergence of the functional series $\sum_{x\in X} e^{-s d(x, z)} \mathcal{P}_x$ in $L^2(\partial M)$, see Proposition \ref{prop-positive-uncond} in the Appendix.
For any $w\in M$ we have $0 \le e^{-s d(x, w)} \mathcal{P}_x \le e^{s d(w, z)}  e^{-s d(x,z)}\mathcal{P}_x$, and Proposition \ref{prop-uncond-cv} follows.
\end{proof}

\subsubsection{Step 3} We obtain the estimate \eqref{up-exp} for the variance of the linear statistics \eqref{def-pv-M}.
 \begin{proposition}\label{prop-var-up-bd-M}
For any $z\in M$, there exists a constant $C_z > 0$ such that for any $s > h_M$ we have
\begin{multline}\label{good-L2-M}
   \left \| \sideset{}{^{v.p.}}\sum_{x\in \X}  e^{-s d(x, z)} \mathcal{P}_x \right\|_{L^2(\Pi; L^2(\partial M))}^2  \le\\\le \frac{C_z}{1 - e^{- (s-h_M)}} +  \|\mathcal{P}_z\|_{L^2(\partial M)}^2  \left(\int_{M}   e^{-s d(x, z)} d\mu_M(x) \right)^2;
\end{multline}
\begin{equation}\label{up-exp}
\Var_\Pi  \left ( \sideset{}{^{v.p.}}\sum_{x\in \X}  e^{-s d(x, z)} \mathcal{P}_x \right) \le  \frac{C_z}{1 - e^{- (s-h_M)}}.
\end{equation}\end{proposition}
\begin{proof}
Corollary \ref{cor-L1-sum-M} and Corollary \ref{cor-MVP} imply,
for any $s> h_M$ and any $z\in M$,  the equality
\begin{equation}\label{lem-av-pv-M}
\E_\Pi \left(  \sideset{}{^{v.p.}}\sum_{x\in \X}  e^{-s d(x, z)} \mathcal{P}_x \right) =  \mathcal{P}_z \cdot \E_\Pi \left(  \sum_{x\in \X}  e^{-s d(x, z)} \right)  = \mathcal{P}_z \cdot \int_M e^{-s d(x,z)} d\mu_M(x).
\end{equation}
For any positive integer $N$, by Proposition \ref{rem-scalar-vector} and  \eqref{lem-av-pv-M},  we have
\begin{multline*}
  \left \| \sum_{k = 0}^{N-1}\sum_{x\in \X \atop k \le d(x, z) < k+1}  e^{-s d(x, z)} \mathcal{P}_x \right\|_{L^2(\Pi; L^2(\partial M))}^2  =   \left \| \sum_{x\in \X \cap B(z, N)}  e^{-s d(x, z)} \mathcal{P}_x \right\|_{  L^2 (\Pi; L^2(\partial M))}^2 =
\\
 =   \Var_\Pi \left(    \sum_{x\in \X \cap B(z, N)}  e^{-s d(x, z)} \mathcal{P}_x \right)  +   \left \|     \E_\Pi \left( \sum_{x\in \X \cap B(z, N)}  e^{-s d(x, z)} \mathcal{P}_x  \right) \right\|_{L^2(\partial M)}^2\le
\\
\le  C \int_{B(z, N)} e^{-2s d(x, z)} \| \mathcal{P}_x\|_{L^2(\partial M)}^2 d\mu_M(x) + \| \mathcal{P}_z \|_{L^2(\partial M)}^2  \left( \int_{B(z, N)}  e^{-s d(x,z)} d\mu_M(x) \right)^2\le
\\
\le C  \underbrace{\int_M e^{-2s d(x, z)} \| \mathcal{P}_x\|_{L^2(\partial M)}^2 d\mu_M(x)}_{\text{denoted by $I_z(s)$}} + \| \mathcal{P}_z \|_{L^2(\partial M)}^2  \left( \int_M  e^{-s d(x,z)} d\mu_M(x) \right)^2.
\end{multline*}
 Assumption \ref{ass-P-growth} implies
\begin{multline*}
I_z(s) \le    \int_M e^{-2s d(x, z)}  \Theta(d(x, o)) e^{h_M  d(x, o)} d\mu_M(x)
\\
 \le  e^{h_M d(z, o) }\int_M e^{-(2s-h_M) d(x, z)}  \Theta( d(x, o)) d\mu_M(x)
\\ =   e^{h_M d(z, o) } \sum_{k=0}^\infty   \int_{A_k(z)}  e^{-(2s-h_M) d(x, z)}  \Theta( d(x, o)) d\mu_M(x)
\\
\le e^{h_M d(z, o) } \sum_{k=0}^\infty   e^{- (2s - h_M) k }  \Theta( k + 1 + d(z, o)) \int_{A_k(z) }d\mu_M
\\
\le e^{h_M d(z, o) } \sum_{k=0}^\infty   e^{- (2s - h_M) k }  \Theta( k + 1 + d(z, o)) \mu_M  (B(z, k+1)).
\end{multline*}
Using Assumption \ref{ass-M-A1}, we obtain
\begin{multline*}
I_z(s)  \le e^{h_M d(z, o) } \sum_{k=0}^\infty   e^{- (2s - h_M) k }  \Theta( k + 1 + d(z, o)) U_z(k+1) e^{h_M( k+1)}
\\
= e^{h_M (1 +d(z, o) )} \sum_{k=0}^\infty   e^{- 2(s - h_M) k }  \Theta( k + 1 + d(z, o)) U_z(k+1).
\end{multline*}
Since $\Theta$ and $U_z$ are both sub-exponential, there exists $c_z >0$ such that
\[
\Theta( k + 1 + d(z, o)) U_z(k+1) \le c_z e^{(s - h_M)k} \quad \text{for all $k \ge 0$}
\]
and  $C_z> 0$ such that
$
I_s(z)  \le  C_z \sum_{k=0}^\infty e^{- (s-h_M)k} = {C_z}{ (1 - e^{- (s-h_M)})^{-1}}.
$
\end{proof}

\subsubsection{Step 4}
Proposition \ref{prop-uncond-cv} implies that for any $z\in M$, any $s >h_M$ and any configuration $X$ lying in the set  $\Omega_M(z)$ defined in \eqref{def-omega-z}, the
series \eqref{def-pv-M} converges absolutely in $L^2(\partial M)$.
For any $s > h_M, z \in M$ and $X\in \Omega_M(z)$, set
\begin{align}\label{def-notation-sigma}
\begin{split}
 &  \Sigma(z, s; X):  = \sideset{}{^{v.p.}}\sum_{x \in X}  e^{-s d(x, z)} \mathcal{P}_x  =  \sum_{x \in X}  e^{-s d(x, z)} \mathcal{P}_x,
\\
& \sigma(z, s; X):  = \sum_{x \in X}  e^{-s d(x, z)},   \quad \overline{\sigma}(z, s) :=  \E_\Pi \Big( \sum_{x \in \X}  e^{-s d(x, z)} \Big),
\\
  &  R(z, s; X) : = \frac{\Sigma(z, s; X)}{\sigma(z, s; X)},  \quad  \underline{R}(z, s; X) : = \frac{\Sigma(z, s; X)}{\overline{\sigma}(z, s)}.
\end{split}
\end{align}
Take  a countable dense subset $\mathcal{A} \subset M$, a sequence $\mathfrak{s} = (s_n)_{n\ge 1}$ in $(h_M, \infty)$ such that $\sum_{n=1}^\infty (s_n - h_M) <\infty$, and introduce a measurable subset $\widehat{\Omega}_M (\mathfrak{s}, \mathcal{A}) \subset \Conf(M)$ by the formula
\begin{align}\label{const-ae-set-M}
\widehat{\Omega}_M (\mathfrak{s}, \mathcal{A}): =  \bigcap_{z \in \mathcal{A}} \left\{  X\in \Omega_M(z):  \sum_{n=1}^\infty \Big\| \underline{R}(z, s_n; X) - \mathcal{P}_{z} \Big\|_{L^2(\partial M)}^2 <\infty \right\}.
\end{align}
By definition, we have $\Pi(\widehat{\Omega}_M (\mathfrak{s}, \mathcal{A})) = 1$.
\begin{proposition}\label{thm-subseq-M}
For any $X \in \widehat{\Omega}_M (\mathfrak{s}, \mathcal{A})$ and any $z\in M$, we have
\begin{align}\label{thm-subseq-M-goal}
\lim_{n\to\infty} \left\|  \underline{R}(z, s_n; X)  - \mathcal{P}_{z} \right\|_{L^2(\partial M)} = 0;
\end{align}
\begin{align}\label{thm-subseq-M-goal-bis}
\lim_{n\to\infty} \left\|  R(z, s_n; X)  - \mathcal{P}_{z} \right\|_{L^2(\partial M)} = 0.
\end{align}
\end{proposition}

The main point here is that  \eqref{thm-subseq-M-goal},  \eqref{thm-subseq-M-goal-bis} hold for all $z\in M$.
Theorem \ref{thm-rec-M} follows from \eqref{thm-subseq-M-goal-bis}.  We  prepare a simple
\begin{lemma}\label{lem-lower-bdd}
For any $z\in M$, we have
\begin{align}\label{lem-lower-bdd-goal}
\liminf_{s \to h_M^{+}}  \left( (s - h_M) \overline{\sigma}(z,  s) \right) > 0.
\end{align}
\end{lemma}
\begin{proof}
By Assumption \ref{ass-M-A2},  there exist $R>0, \delta > 0$ such that $\mu_M(B(o, r)) \ge  \delta e^{r \cdot h_M }$ for any $r \ge R$.
Write
\begin{align}\label{int-rep-exp}
e^{-s t} = \int_{\R^{+}}   s e^{-s r}  \cdot \mathds{1}(t < r) dr,
\end{align}
whence
\begin{multline*}
\overline{\sigma}(z,  s)=\E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(z, x)}\Big]  \ge   e^{-s d(o, z)}  \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big]
\ge e^{-s d(o, z)}  \int_{B(o, R)^c} e^{-s d(o, x)}d\mu_M(x) \ge
\\
 \ge       e^{-s d(o, z)}   \int_{B(o, R)^c} \Big(\int_{\R^{+}}  s e^{- s r} \mathds{1} ( d(o, x) < r) dr \Big) d\mu_M(x)\ge \\
  \ge e^{-s d(o, z)}\int_{R}^\infty s e^{-s r} [\mu_M( B(o, r)) - \mu_M(B(o, R))] dr\ge
 \\
\ge   \delta e^{-s d(o, z)} \int_{R}^\infty s e^{-(s-h_M) r} dr -    \mu_M(B(o, R)) e^{-s d(o, z) -s R} =
\\
=  \frac{\delta s   e^{-s d(o, z)} e^{- (s-h_M)R}}{s-h_M}  -    \mu_M(B(o, R)) e^{-s d(o, z) -s R}.
\end{multline*}
\end{proof}

\begin{proof}[Proof of Proposition \ref{thm-subseq-M}]
Since
$
\lim\limits_{s\to h_M^+} {(s-h_M)}/{(1 - e^{- (s- h_M)})}=1,
$
using Proposition \ref{prop-var-up-bd-M}, Lemma \ref{lem-av-pv-M} and Lemma \ref{lem-lower-bdd}, from the definition \eqref{def-notation-sigma} we obtain
\[
\E_\Pi\left( \sum_{n=1}^\infty \Big\| \underline{R}(z, s_n; \X) - \mathcal{P}_{z} \Big\|_{L^2(\partial M)}^2\right)<\infty.
\]
Now let $X_0 \in \widehat{\Omega}_M (\mathfrak{s}, \mathcal{A})$ and  $z_0\in M$ be fixed. Write
$
\underline{R}(z, s): = \underline{R}(z, s; X_0).
$
  Since the subset $\mathcal{A}\subset M$ is dense, there exists a sequence $(z^{(k)})_{k\ge 1}$ in $\mathcal{A}$, such that $\lim_{k\to\infty} d(z^{(k)}, z_0) =0$. By the definition \eqref{const-ae-set-M} of $\widehat{\Omega}_M(\frak{s}, \mathcal{A})$, since $z^{(k)}$ are points in $\mathcal{A}$, for any $k$, we have
\begin{align}\label{A-ok}
\lim_{n\to\infty} \left\|  \underline{R}(z^{(k)}, s_n)  - \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)} = 0.
\end{align}
Since $\mathcal{P}_x\ge 0$, for any $\xi \in \partial M$, $k, n\in \N$ we have
\[
e^{-s_n d(z_0, z^{(k)})}\sum_{x\in X_0} e^{-s_n d(x, z^{(k)})} \mathcal{P}_x(\xi)  \le \sum_{x\in X_0} e^{-s_n d(x, z_0)} \mathcal{P}_x(\xi)  \le  e^{s_n d(z_0, z^{(k)})} \sum_{x\in X_0} e^{-s_n d(x, z^{(k)})} \mathcal{P}_x(\xi),
\]
whence the following inequalities of  $L^2(\partial M)$-functions:
\[
e^{-s_n d(z_0, z^{(k)})} \sideset{}{^{v.p.}}\sum_{x\in X_0}  e^{-s_n d(x, z^{(k)})}  \mathcal{P}_x \le \sideset{}{^{v.p.}}\sum_{x\in X_0}  e^{-s_n d(x, z_0)}  \mathcal{P}_x \le e^{-s_n d(z_0, z^{(k)}}  \sideset{}{^{v.p.}}\sum_{x\in X_0}  e^{-s_n d(x, z^{(k)})}  \mathcal{P}_x.
\]
Similar inequalities hold for $\overline{\sigma}(z_0, s_n)$ and $\overline{\sigma}(z^{(k)}, s_n)$:
\[
e^{-s_n d(z_0, z^{(k)})}\overline{\sigma}(z^{(k)}, s_n) \le \overline{\sigma}(z_0, s_n) \le e^{s d(z_0, z^{(k)})}\overline{\sigma}(z^{(k)}, s_n).
\]
It follows that
\[
e^{-2 s_n d(z_0, z^{(k)})}\underline{R}(z^{(k)}, s_n) \le  \underline{R}(z_0, s_n)  \le   e^{2 s_n d(z_0, z^{(k)})} \underline{R}(z^{(k)}, s_n),
\]
whence, for any $n, k\in \N$, we obtain
\begin{multline*}
\left\|  \underline{R}(z_0, s_n)  - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}  \le \max_{\pm} \left\|  e^{\pm 2  s_n d(z_0, z^{(k)})}  \underline{R}(z^{(k)}, s_n)  - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}
\\
\le  \max_{\pm} \left\|  e^{\pm 2  s_n d(z_0, z^{(k)})} \underline{R}(z^{(k)}, s_n)  - \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}  + \left\|  \mathcal{P}_{z^{(k)}} - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}
\\
\le  \max_{\pm} \left\|  e^{\pm 2  s_n d(z_0, z^{(k)})}  \underline{R}(z^{(k)}, s_n)  - e^{\pm 2  s_n d(z_0, z^{(k)})}  \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}
\\
 +  \max_{\pm}\left\|   e^{\pm 2  s_n d(z_0, z^{(k)})}  \mathcal{P}_{z^{(k)}} - \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}
  + \left\|  \mathcal{P}_{z^{(k)}} - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}
\\
\le    e^{ 2  s_n d(z_0, z^{(k)})}  \left\|  \underline{R}(z^{(k)}, s_n)  -  \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}
\\
 +  \max_{\pm} |  e^{\pm 2  s_n d(z_0, z^{(k)})}  -1 | \cdot\left\|   \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}
  + \left\|  \mathcal{P}_{z^{(k)}} - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}.
\end{multline*}
Applying \eqref{A-ok} gives
\begin{multline*}
\limsup_{n\to\infty}\left\| \underline{R}(z_0, s_n)  - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}
\\
\le \max_{\pm} |  e^{\pm 2  h_M d(z_0, z^{(k)})}  -1 | \cdot\left\|   \mathcal{P}_{z^{(k)}} \right\|_{L^2(\partial M)}
  + \left\|  \mathcal{P}_{z^{(k)}} - \mathcal{P}_{z_0} \right\|_{L^2(\partial M)}.
\end{multline*}
The continuity of the Busemann cocycle implies that the
map $M \ni z \mapsto \mathcal{P}_z \in L^2(\partial M)$ is continuous.
Since $z^{(k)}\to z_0$  as $k\to\infty$, we  obtain  \eqref{thm-subseq-M-goal}.
It follows that for any $g\in L^2(\partial M)$, we have
\[
\lim_{n\to\infty} \langle g, \underline{R}(z, s_n) - \mathcal{P}_z \rangle_{L^2(\partial M)} = 0,
\]
Taking $g \equiv 1$ gives
\[
\lim_{n\to\infty}  \frac{\sigma(z, s_n; X_0)}{\overline{\sigma}(z, s_n)} =1.
\]
Since
\[
R(z, s_n) =   \underline{R}(z, s_n) \frac{\overline{\sigma}(z, s_n)}{\sigma(z, s_n; X_0)},
\]
the relation  \eqref{thm-subseq-M-goal} implies \eqref{thm-subseq-M-goal-bis}.
\end{proof}

\subsection{Proof of Theorem \ref{thm-rec-M-bis}}
Let  $(M, d, \mu_M)$ be a triple  satisfying Assumptions \ref{ass-M-A2}, \ref{ass-MVP}, \ref{ass-P-growth}, \ref{ass-M-pvg} and let $\Pi$ be a point process on $M$ having  first intensity measure $\mu_M$ and satisfying Assumption \ref{ass-pp}.
Introduce a sequence  $\mathfrak{s}_0 =(s_n)_{n\geq 1}$  by the formula
\begin{equation}\label{sn-def-sq}
s_n= h_M+n^{-2}.
\end{equation}
Using Assumption \ref{ass-M-pvg}, we check that the expectations $\overline{\sigma}(z, s_n)$  defined in \eqref{def-notation-sigma} satisfy
\begin{align}\label{ratio-1-seq}
\lim_{n\to\infty}  \frac{\overline{\sigma}(z, s_n)}{\overline{\sigma}(z, s_{n+1})} = 1.
\end{align}
Using \eqref{ratio-1-seq}, we bound  $R(z, s; X)$ by $R(z, s_n; X)$  and pass from \eqref{thm-rec-M-goal}   to \eqref{thm-rec-M-bis-goal}.

\begin{lemma}\label{lem-R-1-seq}
For any $z\in M$, the sequence $s_n$ given by \eqref{sn-def-sq} satisfies \eqref{ratio-1-seq}.
\end{lemma}

\begin{proof}
Set
\[
\kappa_z(r) : = \frac{\mu_M(B(z, r))}{c_z r^\beta e^{rh_M}}.
\]
By Assumption \ref{ass-M-pvg}, $\lim_{r\to\infty} \kappa_z(r) = 1$. From \eqref{int-rep-exp} we obtain
\begin{multline*}
\int_M e^{-s_n d(x, z)} d\mu_M(x)  = \int_M d\mu_M(x)\int_{\R^{+}}   s_n e^{-s_n r}  \cdot \mathds{1}( d(x, z) < r) dr
\\
 =  s_n \int_{\R^{+}} e^{- s_n r}  \mu_M( B(z, r)) dr = c_z s_n\int_{\R^{+}}   e^{-  r/n^2}  r^\beta \kappa_z(r) dr = c_z   n^{2 + 2 \beta} s_n \int_{\R^{+}}  e^{- t} t^\beta \kappa_z(n^2 t) dt,
\end{multline*}
and the Dominated Convergence Theorem gives
\begin{multline*}
\lim\limits_{n\to\infty}  \frac{\overline{\sigma}(z, 1/n^2 + h_M)}{\overline{\sigma}(z, 1/(n+1)^2 + h_M)}=
\lim\limits_{n\to\infty} \frac{\displaystyle \int_M e^{-s_n d(x, z)} d\mu_M(x) }{\displaystyle  \int_M e^{-s_{n+1}d(x, z)} d\mu_M(x)}  =\\= \lim_{n\to\infty} \frac{ n^{2 + 2 \beta} s_n}{ (n+1)^{2  + 2 \beta} s_{n+1}} \cdot  \frac{\displaystyle \int_{\R^{+}} e^{-t} t^\beta \kappa_z (n^2t ) dt}{\displaystyle \int_{\R^{+}} e^{-t} t^\beta \kappa_z ( (n+1)^2 t) dt }
 = 1.
\end{multline*}
\end{proof}

Recall that to a countable dense subset $\mathcal{A} \subset M$ we have assigned in \eqref{const-ae-set-M} a
Borel subset  $\widehat{\Omega}_M( \mathfrak{s}_0, \mathcal{A})\subset \Conf(M)$.  Theorem  \ref{thm-rec-M-bis} follows from
\begin{proposition}\label{prop-rec-M-bis}
For any countable dense subset $\mathcal{A} \subset M$, any $X\in \widehat{\Omega}(\mathfrak{s}_0, \mathcal{A})$ and any $z\in M$, we have
\begin{align}\label{prop-rec-M-goal-bis}
\lim_{s \to \infty} \left\|  R(z, s; X)  - \mathcal{P}_{z} \right\|_{L^2(\partial M)} = 0.
\end{align}
\end{proposition}

\begin{proof}
Fix $X\in \widehat{\Omega}(\mathfrak{s}_0, \mathcal{A})$ and  $z\in M$. We start with the limit equality \eqref{thm-subseq-M-goal} obtained in \S \ref{sec-proof-thm-rec-M}:
\begin{align}\label{seq-limit-holds}
\lim_{n\to\infty} \| \underline{R}(z, n^{-2} +h_M; X) - \mathcal{P}_z\|_{L^2(\partial M)} =0.
\end{align}
Let us first show that
\begin{align}\label{underline-limit}
\lim_{s\to h_M^{+}} \| \underline{R}(z, s; X) - \mathcal{P}_z\|_{L^2(\partial M)} =0.
\end{align}
From now on, we only consider $s \in (h_M, h_M +1)$. For such $s$, there exists a unique  $n_s \in \N$ such that
$
(n_{s}+1)^{-2} + h_M \le s < n_s^{-2} + h_M.
$
The positivity of the function $\mathcal{P}_x$ and the monotonicity of the function $s \mapsto   e^{- s d(x, z)}$ for any fixed $x, z\in M$ imply
\begin{align}\label{2-side-control-ratio}
\frac{ \Sigma(z, 1/n_s^2 + h_M; X)}{ \overline{\sigma} (z, 1/(n_s+1)^2 + h_M)}  \le \frac{  \Sigma(z, s; X) }{ \overline{\sigma}(z, s)}  \le \frac{ \Sigma(z, 1/(n_s+1)^2 + h_M; X)}{ \overline{\sigma} (z, 1/n_s^2 + h_M)} .
\end{align}
For brevity write
\begin{align*}
\underline{R}^{-}(z, s; X):   =  \underline{R}(z, \, n_s^{-2} + h_M; X),
 & \quad
\underline{R}^{+}(z, s; X):  = \underline{R}(z, \, (n_s+1)^{-2} + h_M; X),
\\
\beta_s:  = & \frac{\overline{\sigma}((n_s+1)^{-2} + h_M)}{\overline{\sigma}(n_s^{-2} + h_M)}.
\end{align*}
Then \eqref{2-side-control-ratio} can be rewritten as
\begin{align*}
 \underline{R}^{-}(z, s; X) \cdot \beta_s^{-1} \le \underline{R}(z, s; X)  \le  \underline{R}^{+}(z, s; X) \cdot \beta_s.
\end{align*}
It follows that
\begin{multline*}\label{2-side-control-simple}
 \| \underline{R}(z, s; X) -\mathcal{P}_z\|_{L^2(\partial M)}
  \le   \max_{\pm}  \| \underline{R}^{\pm}(z, s; X) \beta_s^{\pm 1}  - \mathcal{P}_{z}\|_{L^2(\partial M)}
\\
\le  \max_{\pm}  \| \underline{R}^{\pm}(z, s; X) \beta_s^{\pm 1} - \beta_s^{\pm 1}  \mathcal{P}_{z}\|_{L^2(\partial M)} +  \max_{\pm} | \beta_s^{\pm 1}  -1 | \cdot  \|\mathcal{P}_{z}\|_{L^2(\partial M)}.
\end{multline*}
Lemma \ref{lem-R-1-seq} implies that
$
\lim_{s \to h_M^{+}} \beta_s = 1,
$
while \eqref{seq-limit-holds} implies
$$
\lim_{s\to h_M^{+}} \|\underline{R}^{\pm} (z, s; X) - \mathcal{P}_{z}\|_{L^2(\partial M)} =0,
$$
and  \eqref{underline-limit} is proved.
Taking the expectation in \eqref{underline-limit} gives
\[
\lim_{s \to h_M^{+}}  \frac{\sigma(z, s; X)}{\overline{\sigma}(z, s)} =1,
\]
whence, using the relation
\[
R(z, s; X) =   \underline{R}(z, s; X) \frac{\overline{\sigma}(z, s)}{\sigma(z, s; X)},
\]
we finally obtain  \eqref{prop-rec-M-goal-bis}.
\end{proof}

\section{The reconstruction of $\MM$-harmonic functions}\label{sec-comp-hyper}

 We apply the results of the preceding section to the triple $(\D_d, d_B, \mu_{\D_d})$ consisting of the ball $\D_d$ equipped with the Bergman metric $d_B(\cdot, \cdot)$ given by \eqref{Berg-metric-def} and the associated Bergman volume measure $\mu_{\D_d}$ given by \eqref{Berg-vol-def}.  We choose the origin $0\in \D_d$ as the base point.  Recall,  cf. e.g. \cite[Theorem 2.2.5]{Rudin-ball}, that any biholomorphism
 $\psi \in \Aut(\D_d)$ has the form $\psi = U\varphi_w$, with $U$ a unitary transformation of $\C^d$ and $\varphi_w$ the involution \eqref{inv-auto} .

\subsection{Proof of Theorem \ref{thm-intro-hardy-ball}.}
 We check Assumptions \ref{ass-M-A2}, \ref{ass-MVP},  \ref{ass-P-growth} and \ref{ass-M-pvg} for the triple $(\D_d, d_B, \mu_{\D_d})$.
 For any $z\in \D_d$ and any $r>0$, we have
\begin{align*}
\mu_{\D_d} (B(z, r))  = \mu_{\D_d} (B(0, r))  = \int_{B(0, r)} \frac{dV(z)}{(1 - |z|^2)^{d+1}}=  c_d'\int_{0}^{\frac{e^r - 1}{e^r +1}}  \frac{ t^{2d -1}dt}{(1 - t^2)^{d+1}},
\end{align*}
where $c_d'>0$ is a constant depending only on $d$.  By change of variables $t = \frac{e^x -1}{e^x+1}$, we obtain
\[
\mu_{\D_d} (B(z, r)) = c_d'' \int_{0}^r   e^{-d x} (e^x -1)^{2d-1} (e^x +1) dx,
\]
where $c_d''>0$ is another constant depending only on $d$.
By l'H\^opital principle, we have
\begin{align}\label{pvg-comp-ball}
\lim_{r\to\infty} \frac{\mu_{\D_d}(B(z,r))}{ e^{rd}} =  \lim_{r\to\infty} \frac{ c_d'' e^{-dr} (e^r-1)^{2d -1} (e^r+1)}{d e^{rd}} = \frac{c_d''}{d}> 0.
\end{align}
Therefore, Assumption \ref{ass-M-pvg} is checked. In particular, we have shown that the volume entropy $h_{\D_d} = d$.

There exist $c, C >0$, such that for any $s> d$ and any $z\in \D_d$, we have
\begin{align}\label{lem-conf-inv-2-side-exp-goal}
\frac{c}{s-d} \le \int_{\D_d} e^{-s d_B(x, z)} d\mu_{\D_d} (x) \le \frac{C}{s-d}.
\end{align}
Indeed, sending $z$ to $0$ gives
\begin{align}\label{pol-est-dd}
\int_{\D_d} e^{-s d_B(x, z)} d\mu_{\D_d} (x)    = \int_{\D_d} e^{-s d_B(x, 0)} d\mu_{\D_d} (x)   = \int_{\D_d} \left( \frac{1 - |x|}{1 + |x|} \right)^s  \frac{dV(x)}{( 1 - |x|^2)^{d+1}},
\end{align}
and integrating \eqref{pol-est-dd} in polar coordinates  gives \eqref{lem-conf-inv-2-side-exp-goal}.
Consider the closed unit ball $\overline{\D}_d = \{z\in \C^d: |z|\le 1 \}$ and, as before, endow the unit sphere with the normalized Lebesgue measure
$\sigma_{\Sph_d}$. Set
\[
d\nu_s (z) =  \frac{ e^{-s d_B(z, 0)} d\mu_{\D_d}(z)}{ \displaystyle \int_M e^{-s d_B(z, 0)} d\mu_{\D_d}(z)}.
\]
The bound \eqref{lem-conf-inv-2-side-exp-goal} implies that any accumulation point of $\nu_s$ as $s\to d^+$ is supported on the boundary $\partial \D_d$.
 Since $\nu_s$ is rotationally invariant for any $s>d$, we have
\begin{align}\label{nu-s-to-haar}
\lim\limits_{s\to d^+} \nu_s= \sigma_{\Sph_d}.
\end{align}
 Hence the Assumption \ref{ass-M-A2} is checked.

The Busemann cocyle of the complex hyperbolic space $\D_d$, cf. e.g. \cite[Theorem 2.2.2]{Rudin-ball}, is given , for
 $w, z\in \D_d, \zeta \in \Sph_d$, by the formula
\begin{equation*}
B_\zeta (w, z) =\log \left(  \frac{(1 - |z|^2) \cdot | 1 - \bar{w} \cdot \zeta|^2}{ (1 - |w|^2) \cdot |1 - \bar{z} \cdot \zeta|^2}\right).
\end{equation*}
The Poisson-Szeg\"o kernel $P^b(w, \zeta)$ satisfies
\[
e^{-h_{\D_d} B_\zeta (w, 0)} =    \frac{(1 - |w|^2)^d}{ | 1 - \bar{w}\cdot \zeta|^{2d}}=P^b(w, \zeta),
\]
and coincides, for the complex hyperbolic space, with the kernel defined by the formula  \eqref{def-abstract-P-kernel}.
By definition \eqref{def-abstract-hardy}, we have
\begin{align}\label{H-P-b-MMHH}
\HH^2_{P^b} (\D_d) = \MM\HH^2(\D_d).
\end{align}

 By \cite[Theorem 3.3.7]{Rudin-ball}, for any fixed $\zeta\in \Sph_d$,  the continuous function $ w \mapsto  P^b(w, \zeta)$ satisfies the invariant mean value property
\begin{align}\label{inv-MVP}
P^b (w, \zeta)  = \int_{\Sph_d} P^b (\varphi_w(r \xi), \zeta ) d\sigma_{\Sph_d} (\xi) \quad \text{for all $w \in \D_d$ and $0 < r <1$.}
\end{align}
Let $m_{\mathcal{U}_d}$ denote the normalized Haar measure on the unitary group $\mathcal{U}_d$. Since $\sigma_{\Sph_d}$ coincides with the orbital measure of the transitive action of $\mathcal{U}_d$ on $\Sph_d$, the equality \eqref{inv-MVP} is equivalent to
\begin{align}\label{inv-MVP-2}
P^b (w, \zeta) =  \int_{\mathcal{U}_d} P^b ( \varphi_w(U x ),  \zeta ) dm_{\mathcal{U}_d} (U) \quad \text{for any $w, x \in \D_d$},
\end{align}
whence, for any $z\in \D_d, \zeta \in \Sph_d$ and $r > 0$, we have
\begin{align*}
P^b(z, \zeta)  = \frac{1}{\mu_{\D_d} ( B(z, r))} \int_{B(z, r)} P^b( x, \zeta) d\mu_{\D_d}(x).
\end{align*}
Hence Assumption \ref{ass-MVP} is checked.

Proposition 1.4.10 in \cite{Rudin-ball} implies there exists $c>0$ such that
\[
\int_{\Sph_d} P^b(x, \zeta)^2 d \sigma_{\Sph_d}(\zeta) \le  \frac{c}{(1 - |x|^2)^d}.
\]
Since $e^{h_{\D_d} d(x, 0)} = ( 1 + |x|)^d ( 1 - |x|)^{-d}  \ge (1 - |x|^2)^{-d}$,
there exists a constant $c> 0$  such that
\begin{equation}\label{lem-L2-P-b}
\int_{\Sph_d} P^b(x, \zeta)^2 d \sigma_{\Sph_d}(\zeta) \le   c e^{h_{\D_d} d(x, 0)} \quad \text{for all $x\in \D_d$}.
\end{equation}
Hence Assumption \ref{ass-P-growth} is checked.

We have checked that the triple $(\D_d, d_B, \mu_{\D_d})$ satisfies Assumptions \ref{ass-M-A2}, \ref{ass-MVP},  \ref{ass-P-growth} and \ref{ass-M-pvg}. Therefore,  Theorem \ref{thm-intro-hardy-ball} follows from Theorem \ref{thm-rec-M-bis}.
\qed

\subsection{The reconstruction of a fixed Bergman function}\label{sec-single-M}

Set
\[
b^2(\D_d): = \left\{f \in \MM \HH (\D_d)\Big| \int_{\D_d}  |f(z)|^2 dV(z) <\infty \right\}.
\]
By \cite[Theorem 4.2.4, Corollary 2]{Rudin-ball}, any $\MM\HH$-harmonic function $f\in \MM(\D_d)$ satisfies the invariant mean value property:
\begin{align}\label{MVP-m-harm}
f(w) = \int_{\mathcal{U}_d} f(\varphi_w( Ux )) dm_{\mathcal{U}_d}(U) \quad \text{for all $w, x \in \D_d$}.
\end{align}

\begin{proposition}\label{prop-single-f}
Let $\Pi$ be a point process on $\D_d$ satisfying Assumption \ref{ass-pp} and with first intensity measure $\lambda \mu_{\D_{d}}$ for some constant $\lambda>0$. Fix $f \in b^2(\D_d)$ and $z \in \D_d$. Then
\begin{align}\label{pv-single-f}
\sum_{k=0}^\infty \Big| \sum_{x \in X \atop k \le d_B(x, z) < k+1} e^{-s d_B(z, x)} f(x)\Big|<\infty, \quad \text{for $\Pi$-almost every $X$.}
\end{align}
Moreover, let $\mathfrak{s} = (s_n)_{n\ge 1}$ be a fixed sequence such that  $d < s_n < \infty$ and $\sum_{n=1}^\infty (s_n-d)^2 <\infty$.
Then there exists a subset $\widehat{\Omega}_{\mathfrak{s}, z, f} \subset \Conf(\D_d)$ with $\Pi (\widehat{\Omega}_{\mathfrak{s}, z, f} )= 1$ such that  for any $X\in \widehat{\Omega}_{\mathfrak{s}, z, f}$, we have
\begin{align}\label{reconst-single}
 f(z) = \lim_{n\to\infty} \frac{\displaystyle{\sum_{k=0}^\infty \sum_{x \in X \atop k\le d_B(z,x) < k+1} e^{-s_n d_B(z, x)} f(x)}}{\displaystyle{  \sum_{x \in X} e^{-s_n d_B(z,x)} }}.
\end{align}
\end{proposition}

\begin{proof}
Fix $f \in b^2(\D_d)$ and $z \in \D_d$. To prove \eqref{pv-single-f}, it suffices to prove that
\begin{align}\label{L-1-2-single}
\ \sum_{k=0}^\infty \Big\| \underbrace{ \sum_{x \in \X \atop k \le d_B(x, z) < k+1} e^{-s d_B(z, x)} f(x)}_{\text{denoted by $T_k(s, \X)$}}\Big\|_{L^2(\Pi)}<\infty.
\end{align}
 For any $k\in \mathbb N$, we have
\begin{align}\label{L-2-single-f}
\Big\|  T_k(s, \X) \Big\|_{L^2(\Pi)}^2
=\Var_\Pi  \Big(  T_k(s, \X) \Big)  + \left|  \E_\Pi \left(   T_k(s, \X) \right) \right|^2.
\end{align}
Set $A_k(z) = \{x\in \D_d: d_B(z, x) \in [k, k+1)\}$.
 By the formula \eqref{Berg-metric-def} for $d_B(\cdot, \cdot)$ and the  identity (see e.g. \cite[Theorem 2.2.2]{Rudin-ball})
\[
 \frac{1}{1 - | \varphi_{z}(x)|^2} = \frac{|  1- \bar{x} \cdot z|^2}{(1 - |z|^2) (1 - |x|^2)} \quad x, z \in \D_d,
\]
there exists $C_z>0$, such that for any $x\in A_k(z)$, we have $(1 - |x|^2)^{-1} \le C_z e^{k}$.  Then by Assumption \ref{ass-pp}, we obtain
\begin{multline*}
\Var_\Pi  \left(   T_k(s, \X)  \right)   \le  C \E_\Pi  \Big(  \sum_{x \in \X \atop k \le d_B(z,x) < k+1} e^{-2s d_B(z, x)} |f(x)|^2  \Big)
\\
 =  C \lambda \int_{A_k(z)}e^{-2s d_B(z, x)} | f(x)|^2 \frac{dV(x)}{(1 - |x|^2)^{d+1}}
\\
\le  C\lambda \max_{x\in A_k(z)} \frac{e^{-2s d_B(z, x)}}{( 1 - |x|^2)^{d+1}} \int_{\D_d} | f(x)|^2 dV(x)  \le C' e^{-(2s-d -1)k},
\end{multline*}
where $C'>0$ is a constant depending on $f, s, z, \Pi$. Using the invariance of the measure $\mu_{\D_d}$ by biholomorphic automorphisms of $\D_d$,  the rotational invariance of the measure $e^{-s d_B(0, x)} d\mu_{\D_d}(x)$ and the equality \eqref{MVP-m-harm}, we obtain
\begin{align}\label{exp-T-s-X}
\begin{split}
 \E_\Pi \left(  T_k (s, \X)\right) =  \lambda \int_{A_k(z)} e^{-s d_B(z, x)} f(x)  d\mu_{\D_d} (x)
=  \lambda \int_{A_k(0)} e^{-s d_B(0, x)}  f(\varphi_z(x))  d\mu_{\D_d}(x)
\\
=  \lambda  \int_{\mathcal{U}_d} d m_{\mathcal{U}_d}(U) \int_{A_k(0)} e^{-s d_B(0, x)}  f(\varphi_z( U x))  d\mu_{\D_d}(x) =  \lambda f(z) \int_{A_k(0)} e^{-s d_B(0, x)}  d\mu_{\D_d}(x).
\end{split}
\end{align}
Substituting  into \eqref{L-2-single-f}, we obtain
\[
\left\|  T_k (s, \X) \right\|_{L^2(\Pi)} \le C'' \Big(   e^{-(2 s - d -1)k/2} +  \int_{A_k(0)} e^{-s d_B(0, x)}   d\mu_{\D_d}(x) \Big),
\]
where $C''>0$ is a constant depending on $f, s, z, \Pi$. If $s>d$, then $2s - d -1> 0$, hence by \eqref{lem-conf-inv-2-side-exp-goal}, we have
\begin{align}\label{L2-summable-single-f}
\sum_{k=0}^\infty \left\| T_k(s, \X) \right\|_{L^2(\Pi)} \le C'' \Big(    \sum_{k=0}^\infty e^{-(2s-d -1)k/2} +  \int_{\D_d} e^{-s d_B(0, x)}  d \mu_{\D_d}(x) \Big) <\infty.
\end{align}

Our next step is to show that
\begin{align}\label{var-single}
\sup_{s \in (d, \infty) }\Var_\Pi \Big( \sum_{k=0}^\infty  T_k(s, \X) \Big) < \infty.
\end{align}
Note that
\[
C'''_z = \sup_{s\in (d, \infty)} \sup_{x\in \D_d} \frac{ e^{- 2 s d_B(z, x)}}{ ( 1 - |x|^2)^{d+1}}   <\infty.
\]
For $N\in \N$ and  $s > d$ we have
\begin{multline*}
 \Var_\Pi \Big( \sum_{k=0}^{N-1}  T_k (s, \X) \Big)  = \Var_\Pi \Big(  \sum_{x \in \X, \,  d_B (z,x) < N} e^{-s d_B(z, x)} f(x)\Big)
\\
   \le  C  \E_\Pi \Big( \sum_{x \in \X } e^{-2s d_B(z, x)} |f(x)|^2 \Big) = C \lambda \int_{\D_d}  e^{-2s d_B(z, x)} |f(x)|^2 d\mu_{\D_d}(x)
\\
 \le C C'''_z \lambda \int_{\D_d} | f(x)|^2 dV(x).
\end{multline*}
Letting $N \to \infty$, we obtain the desired inequality \eqref{var-single}.

For  $s > d$, by \eqref{L-1-2-single} and \eqref{exp-T-s-X},  we have
\begin{multline*}
 \E_\Pi \Big( \sum_{k=0}^\infty T_k(s, \X) \Big)   =  \sum_{k=0}^\infty \E_\Pi \Big( T_k(s, \X) \Big) = \sum_{k = 0}^\infty \lambda f(z) \int_{A_k(0)} e^{-s d_B(0, x)}  d\mu_{\D_d}(x)
\\
 = \lambda f(z) \int_{\D_d} e^{-s d_B(0, x)}  d\mu_{\D_d}(x) = \lambda f(z) \int_{\D_d} e^{-s d_B(z, x)}  d\mu_{\D_d}(x)
 = f(z) \cdot \E_\Pi \Big( \sum_{x \in \X} e^{-s d_B(z, x)} \Big),
\end{multline*}
which combined  with the inequalities \eqref{var-single}, \eqref{lem-conf-inv-2-side-exp-goal} and  the assumption $\sum_{n=1}^\infty (s_n -d)^2 <\infty$, implies
\[
\sum_{n=1}^\infty \E_\Pi \left(   \left|  \frac{\displaystyle{\sum_{k=0}^\infty    T_k(s_n, \X) }}{ \displaystyle{\E_\Pi \left( \sum_{x \in \mathscr{Y}} e^{-s_n d_B(z, x)} \right)} }   - f(z) \right|^2 \right)<\infty.
\]
Therefore, for $\Pi$-almost every $X$, we have
\[
f(z) = \lim_{n\to\infty}\frac{\displaystyle{\sum_{k=0}^\infty \sum_{x \in X \atop k\le d_B(z,x) < k+1} e^{-s_n d_B(z, x)} f(x)}}{ \displaystyle{\E_\Pi \left( \sum_{x \in \X} e^{-s_n d_B(z, x)} \right)} }.
\]
In particular, taking $f \equiv 1$, we obtain that for $\Pi$-almost every $X$,
\[
\lim_{n\to\infty}\frac{\displaystyle{\sum_{k=0}^\infty \sum_{x \in X \atop k\le d_B (z,x) < k+1} e^{-s_n d_B(z, x)}}}{ \displaystyle{\E_\Pi \left( \sum_{x \in \X} e^{-s_n d(z, x)} \right)} }   = \lim_{n\to\infty} \frac{\displaystyle{\sum_{x \in X} e^{-s_n d_B(z, x)}}}{ \displaystyle{\E_\Pi \left( \sum_{x \in \X} e^{-s_n d_B(z, x)} \right)} }  = 1.
\]
Thus, for $\Pi$-almost every $X$, the desired limit equality \eqref{reconst-single} holds.
\end{proof}

\section{The reconstruction of $\HH$-harmonic functions}\label{sec-real-hyper}

Let $m\ge 2$ be an integer.  We apply the results of \S \ref{sec-general-rec}  to the triple $(\B_m, d_h, \mu_{\B_m})$ consisting of the real hyperbolic space $\B_m$ equipped with the hyperbolic metric $d_h(\cdot, \cdot)$ given by \eqref{def-d-h} and the associated hyperbolic volume measure $\mu_{\B_m}$ given by \eqref{hyper-vol-def}.  We choose the origin $0\in \B_m$ as the base point.  The hyperbolic volume measure $\mu_{\B_m}$ is invariant under the action of the group $\Aut(\B_m)$ of all M\"obius transformations preserving  $\B_m$. Any  $\phi \in \Aut(\B_m)$ has the form $ \phi  = A \psi_a$, where $A$ is an orthogonal transformation of $\R^m$ and $\psi_a$ is the involution defined in \eqref{def-psi-a}.

\subsection{ Proof of  Theorem \ref{thm-intro-hyperball}}
 We check Assumptions \ref{ass-M-A2}, \ref{ass-MVP},  \ref{ass-P-growth} and \ref{ass-M-pvg} for the triple $(\B_m, d_h, \mu_{\B_m})$.

By the same argument for proving the limit equality \eqref{pvg-comp-ball}, we show that  there exists $c_m>0$, such that for any $a\in \B_m$, we have
\[
\lim_{r\to\infty} \frac{\mu_{\B_m} (B(a, r))}{  c_m e^{r (m-1)}}  = 1.
\]
Therefore, Assumption \ref{ass-M-pvg} is checked. In particular, we have shown that the volume entropy $h_{\B_m} = m-1$.

By the same argument for obtaining the weak convergence \eqref{nu-s-to-haar}, we show that
\[
\lim_{ s \to (m-1)^{+}}  \frac{ e^{-s d_h(x, 0)} d\mu_{\B_m}(x)}{ \displaystyle \int_M e^{-s d_h(x, 0)} d\mu_{\B_m}(x)}  =  \sigma_{S^{m-1}} (d\zeta).
\]
Hence the Assumption \ref{ass-M-A2} is checked.

The Busemann cocyle of the real hyperbolic space $\B_m$, cf.\cite[Formula (2.1.7)]{Stoll-hyper-ball}, is given, for $x, y \in \B_m, t \in S^{m-1}$, by the formula
\begin{align*}
B_t (x, y)  =\log \left(   \frac{|x - t|^2}{1 - |x|^2} \cdot \frac{| y - t|^2}{1 - |y|^2} \right).
\end{align*}
The hyperbolic Poisson kernel $P^h(x, t)$ satisfies
\[
e^{-h_{\B_m} B_t (x, 0)} =   \left( \frac{(1 - |x|^2)}{ | x  - t|^{2}}\right)^{m-1}=P^h(x, t),
\]
and coincides, for the real hyperbolic space, with the kernel defined by the formula  \eqref{def-abstract-P-kernel}.
By definition \eqref{def-abstract-hardy}, we have
\begin{align}\label{H-P-h-HH}
\HH^2_{P^h} (\B_m) = \HH^2(\B_m).
\end{align}

By \cite[Lemma 5.3.1]{Stoll-hyper-ball}, for any fixed $t \in S^{m-1}$, the continuous function $x \mapsto P^h(x, t)$ is $\HH$-harmonic, thus satisfies
\begin{align}\label{MVP-hyper-P}
P^h(a, t)  = \int_{\mathcal{O}_m} P^h(\psi_a (A x), t) m_{\mathcal{O}_m}(dA)\quad \text{for any $a, x \in \B_m$,}
\end{align}
where $\mathcal{O}_m$ is the group of $m\times m$ orthogonal matrices and $m_{\mathcal{O}_m}$ is the normalized Haar measure on $\mathcal{O}_m$. Therefore,  for any $a\in \B_m, t \in S^{m-1}$ and  $r > 0$, we have
\begin{align*}
P^h(a, t)  = \frac{1}{\mu_{\B_m} ( B(a, r))} \int_{B(a, r)} P^h( x, t) d\mu_{\B_m}(x).
\end{align*}
Hence Assumption \ref{ass-MVP} is checked.

By Theorem 5.5.7 in \cite{Stoll-hyper-ball} implies there exists $c> 0$ such that
\begin{align}\label{2-norm-poi-hyp}
\int_{S^{m-1}} P^h(x, t)^2 d \sigma_{S^{m-1}}(t) \le \frac{c}{(1 - |x|^2)^{m-1}} \quad \text{for all $x\in \B_m$ and any $t\in S^{m-1}$}.
\end{align}
This implies that there exists a constant $c'> 0$  such that
\[
\int_{S^{m-1}} P^b(x, t)^2 d \sigma_{S^{m-1}}(t) \le   c' e^{h_{\B_m} d(x, 0)} \quad \text{for all $x\in \B_m$}.
\]
Hence Assumption \ref{ass-P-growth} is checked.

We have checked that the triple $(\B_m, d_h, \mu_{\B_m})$ satisfies Assumptions \ref{ass-M-A2}, \ref{ass-MVP},  \ref{ass-P-growth} and \ref{ass-M-pvg}. Therefore,  Theorem \ref{thm-intro-hyperball} follows from Theorem \ref{thm-rec-M-bis}.
\qed

\subsection{The reconstruction of a fixed Bergman function}\label{sec-single-H}

Set
\[
b^2(\B_m): = \left\{f \in  \HH (\B_m)\Big| \int_{\B_m}  |f(x)|^2 dV(x) <\infty \right\}.
\]
By  \cite[Corollary 4.1.3]{Stoll-hyper-ball},  any $\HH$-harmonic function $f\in \HH(\B_m)$ satisfies the following mean value property:
\begin{align}\label{MVP-h-harm}
f(a) = \int_{\mathcal{O}_m} f( \psi_a (A x)) m_{\mathcal{O}_m}(dA) \quad \text{for any $a, x \in \B_m$}.
\end{align}

\begin{proposition}\label{prop-single-f-real}
Let $\Pi$ be a point process on $\B_m$ satisfying Assumption \ref{ass-pp} and with first intensity measure $\lambda \mu_{\B_{m}}$ for some constant $\lambda>0$. Fix $f \in b^2(\B_m)$ and $a \in \B_m$. Then
\[
\sum_{k=0}^\infty \Big| \sum_{x \in X \atop k \le d_h(x, a) < k+1} e^{-s d_h(a, x)} f(x)\Big|<\infty, \quad \text{for $\Pi$-almost every $X$.}
\]
Moreover, let $\mathfrak{s} = (s_n)_{n\ge 1}$ be a fixed sequence such that  $m-1< s_n < \infty$ and $\sum_{n=1}^\infty (s_n-m+1)^2 <\infty$.
Then there exists a subset $\widehat{\Omega}_{\mathfrak{s}, z, f} \subset \Conf(\B_m)$ with $\Pi (\widetilde{\Omega}_{\mathfrak{s}, a, f} )= 1$ such that  for any $X\in \widetilde{\Omega}_{\mathfrak{s}, a, f}$, we have
\[
 f(a) = \lim_{n\to\infty} \frac{\displaystyle{\sum_{k=0}^\infty \sum_{x \in X \atop k\le d_B(z,x) < k+1} e^{-s_n d_h(a, x)} f(x)}}{\displaystyle{  \sum_{x \in X} e^{-s_n d_h(a,x)} }}.
\]
\end{proposition}
\begin{proof}
The proof is similar to that of Proposition \ref{prop-single-f}. The r\^ole played by the identity \eqref{MVP-m-harm} is now played by the identity \eqref{MVP-h-harm}.
\end{proof}

\section{Radial weights on the disk}\label{sec-more-disk}

\noindent We return to  the determinantal  point process $\Pi_{K_\D}$ governed by the Bergman kernel $K_{\D}$ on the disk $\D$. We show in  Proposition \ref{thm-reconstruct-hardy} below that, for any fixed point $z\in \D$, the simultaneous reconstruction for all $H^2$-Hardy functions can be achieved using the uniform weights on  hyperbolic disks of growing radius centred at $z$. The reconstruction result  in  Proposition \ref{thm-reconstruct-hardy} is weaker than that in Theorem \ref{thm-disk}:  we only obtain a  limit along a fixed subsequence $s_n\to 1^+$.  In Proposition \ref{prop-failure} we show the impossibility of  simultaneous reconstruction for all functions $f\in A^2(\D)$  using  averaging  with compactly supported radial weights. The argument is based on a new formula for the variance of vector-valued additive statistics, see Proposition \ref{prop-new-var-f}.

\subsection{Reconstruction with compactly supported weights}\label{sec-thm-reconstruct-hardy}

Proposition \ref{thm-reconstruct-hardy} below is a reconstruction result with general compactly supported  radial weights on $\D$.
Recall that the holomorphic Hardy space $H^2(\D)$ is defined by the formula
\begin{align*}
H^2(\D): &= \left\{f: \D\rightarrow \C\Big| \text{$f$ is holomorphic and $ \| f\|_{H^2(\D)}^2 =  \sup_{0< r <1}\int_0^{2 \pi} |f(r e^{i \theta})|^2 \frac{d\theta}{2\pi} <\infty$}\right\}.
\end{align*}
The sequence of monomials $(z^n)$, $n\geq 0$ is an orthonormal basis for the Hardy space $H^2(\D)$,  whose reproducing kernel thus has the form
\[
S(w, x) = (1- \bar{x}w)^{-1}.
\]
In what follows, for any $x \in \D$,  we denote $S_x \in H^2(\D)$  the function given by
\[
S_x (w) = S(w, x) = (1- \bar{x}w)^{-1}.
\]
For $1< s<2$ and $z, x\in \D$, set
\[
W_s(x) =  \left( 1 - |x|^2 \right)^{s} \mathds{1}\left( |x|^2 \le 2 - s\right) \an \varphi_z(x)= (z-x) (1 - \bar{z}x)^{-1}.
\]
 \begin{proposition}\label{thm-reconstruct-hardy}
Let $z\in \D$ be a fixed point and  let  $(s_n)_{n\ge 1}$  be a fixed sequence such that $1 < s_n < 2$  and
\begin{align}\label{log-speed}
\sum_{n=1}^\infty  \frac{1}{| \log (s_n-1)|} <\infty.
\end{align}
 Then for $\Pi_{K_{\D}}$-almost every $X\in \Conf(\D)$, we have
\begin{align}\label{hardy-rep}
\lim_{n\to\infty} \left\|  \frac{\displaystyle \sum_{x\in X} W_{s_n} (\varphi_z(x)) S_x   }{ \displaystyle \sum_{x\in X}  W_{s_n} (\varphi_z(x)) }  - S_{z}  \right\|_{H^2(\D)}  =0
\end{align}
and for any $f\in H^2(\D)$, we have
\begin{align}\label{reconstruct-hardy-eq}
f(z)= \lim_{n\to\infty}  \frac{\displaystyle \sum_{x\in X}  W_{s_n} (\varphi_z(x))  f(x) }{\displaystyle \sum_{x\in X}  W_{s_n} (\varphi_z(x))  } .
\end{align}
\end{proposition}

Unlike Theorem \ref{thm-disk},  Proposition \ref{thm-reconstruct-hardy} claims neither that the formula \eqref{reconstruct-hardy-eq} holds almost surely  for all $f\in H^2(\D)$ simultaneously,  nor that \eqref{reconstruct-hardy-eq} holds  for all $z\in \D$. We do not see how to replace the limit procedure using a fixed sequence $(s_n)$ satisfying \eqref{log-speed} by the limit procedure $s\to 1^{+}$. The
 specific exponential form of the weight $e^{-s d(x,z)}$ plays a key r{\^o}le in obtaining the stronger statements  of Theorem \ref{thm-disk}.

\subsubsection{The variance of linear statistics}
Set
$$
W_s(x) =  \left( 1 - |x|^2 \right)^{s} \mathds{1}\left( |x|^2 \le 2 - s\right)  \an \varphi_z(x) = \frac{z-x}{1 - \bar{z}x}.
$$
The  proof of Proposition \ref{thm-reconstruct-hardy} is based on a formula, cf. Proposition \ref{prop-up-var-weight},   and  a bound,
cf. Corollary \ref{prop-decay-var-hardy},  for the variance
\begin{align}\label{hardy-l-stat}
\Var_{\Pi_{K_\D}} \Big( \sum_{x\in \X}  W_s(\varphi_z(x)) \cdot S_x\Big).
\end{align}
\begin{proposition}\label{prop-up-var-weight}
For any compactly supported radial weight  $W: \D \rightarrow \R^{+}$ and any $z\in \D$, we have
\begin{equation*}
\Var_{\Pi_{K_\D}}\Big[\sum_{x\in  \X} W(\varphi_{z}(x))  S_x \Big]
= \frac{1}{2 \pi^2(1- |z|^2)}\int_\D \int_\D | W(\eta) - W(\xi)|^2  \frac{1 + | z \eta \xi|^2 ( 2- | \eta \xi|^2)}{ (1 - |\eta \xi|^2)^4} dV(\eta) dV(\xi).
\end{equation*}
\end{proposition}
Proposition \ref{prop-up-var-weight} implies
\begin{corollary}\label{prop-decay-var-hardy}
Let $(W_s)_{s\in (1, \frac{5}{4})}$ be the radial weight functions on $\D$ defined by
\[
W_s(x) = ( 1 - | x|^2)^s \mathds{1}(|x|^2 \le 2 - s) \quad x \in \D.
\]
Then for any $z \in \D$, there exists a constant $C_{z}>0$, such that
\begin{align}\label{log-2-decay}
\frac{ \displaystyle  \Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W_s (\varphi_{z}(x)) S_x  \Big]   }{ \displaystyle \Big[   \E_{\Pi_{K_\D}} \Big(    \sum_{x\in  \X} W_s (\varphi_{z}(x))  \Big)   \Big]^2} \le \frac{C_{z}}{\log \Big(\frac{1}{s-1}\Big)}.
\end{align}
\end{corollary}

\subsubsection{Derivation of Proposition \ref{thm-reconstruct-hardy} from Corollary \ref{prop-decay-var-hardy}}
We first show that  for any $z\in \D$ and any $n\ge 1$, we have
\begin{align}\label{eq-radial-vec-weight}
\E_{\Pi_{K_\D}} \left( \sum_{x\in \X} W_{s_n}(\varphi_z(x)) \cdot S_x\right)   = S_z \cdot \E_{\Pi_{K_\D}} \left( \sum_{x\in \X} W_{s_n}(\varphi_z(x))\right) = S_z \cdot \E_{\Pi_{K_\D}} \left( \sum_{x\in \X} W_{s_n}(x)\right).
\end{align}
Indeed, for any $y\in \D$, we have
\begin{align*}
\E_{\Pi_{K_\D}}\left( \sum_{x\in \X}  W_{s_n}\left( \varphi_{z}(x)\right)  S_x(y)\right)  &  = \E_{\Pi_{K_\D}}\left( \sum_{x\in \X}  W_{s_n}(x)  S_{\varphi_{z}(x)} (y)\right)
\\
& = \int_\D W_{s_n}(x) S_{\varphi_{z}(x)}(y) \frac{dV(x)}{ \pi ( 1 - |x|^2)^2}.
\end{align*}
Recalling that  $W$ is radial and the function $x \mapsto S_{x}(y)$ is harmonic on $\D$ for any fixed $y\in \D$ gives
\begin{multline*}
\int_\D W_{s_n}(x) S_{\varphi_{z}(x)}(y) \frac{dV(x)}{ \pi ( 1 - |x|^2)^2} =  \int_{0}^{2 \pi} \frac{d\theta}{2 \pi}\int_\D W_{s_n}(x) S_{\varphi_{z}(x e^{i \theta})}(y) \frac{dV(x)}{ \pi ( 1 - |x|^2)^2}
\\
=   S_{\varphi_z(0)}(y)\int_\D W_{s_n}(x)  \frac{dV(x)}{ \pi ( 1 - |x|^2)^2} =  S_{z}(y) \int_\D W_{s_n}(x)  \frac{dV(x)}{ \pi ( 1 - |x|^2)^2};
\end{multline*}
\begin{align*}
\E_{\Pi_{K_\D}}\left( \sum_{x\in \X}  W_{s_n}\left( \varphi_{z}(x)\right)  S_x(y)\right)    =   S_{z}(y) \int_\D    \frac{ W_{s_n}(x) dV(x)}{ \pi ( 1 - |x|^2)^2}
= S_{z}(y) \int_\D    \frac{W_{s_n}(\varphi_z(x)) dV(x)}{ \pi ( 1 - |x|^2)^2}.
\end{align*}
Since $y \in \D$ is arbitrary, \eqref{eq-radial-vec-weight} follows.
From the equality \eqref{eq-radial-vec-weight} and the estimate \eqref{log-2-decay}, under the assumption \eqref{log-speed}, for any fixed $z\in \D$,  we have
\[
\sum_{n=1}^\infty \E_{\Pi_{K_\D}} \left( \left\|  \frac{ \displaystyle \sum_{x\in \X} W_{s_n} (\varphi_z(x)) \cdot S_x   }{ \displaystyle \E_{\Pi_{K_\D}} \Big( \sum_{x\in \mathscr{Y} }  W_{s_n}(\varphi_z(x))\Big)}  - S_{z} \right\|_{H^2(\D)}^2 \right) <\infty
\]
and \eqref{hardy-rep}. Since for  $f\in H^2(\D)$, $z\in \D$, we have $f(z) = \langle f, S_z\rangle_{H^2(\D)}$, \eqref{reconstruct-hardy-eq} follows.\qed
\subsubsection{Computation of the variance: proof of Proposition \ref{prop-up-var-weight}}
We start with
\begin{proposition}\label{prop-hardy}
If $g\in L^2( \D, dV)$  takes real values and has compact support, then
\begin{align}\label{var-hardy}
\Var_{\Pi_{K_\D}}\Big[     \sum_{x\in  \X} g(x) S_x \Big] = \frac{1}{2} \int_\D \int_\D | g(z) - g(w)|^2 \cdot  S(z, w) |K_\D(z, w)|^2 dV(z) dV(w).
\end{align}
\end{proposition}
Recall the standard Sobolev norm expression  for the variance of linear statistics under determinantal point processes induced by an orthogonal projection.
\begin{lemma}\label{lem-var-sob}
If $\mathscr{H}$  is a Hilbert space and  $f\in L^2( \D, dV; \mathscr{H})$ has compact support, then
\begin{align}\label{f-var}
\Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} f(x)  \Big]= \frac{1}{2} \int_\D \int_\D \|f(z) - f(w) \|_{\mathscr{H}}^2 \cdot  |K_\D(z, w)|^2 dV(z) dV(w).
\end{align}
\end{lemma}
 The key step for obtaining the formula \eqref{var-hardy} is
\begin{lemma}\label{lem-rep-hardy}
For any $z\in \D$ we have
\[
\int_\D S (z, w) |K_\D(z, w)|^2 dV(w) = S(z, z) K_\D(z,z).
\]
\end{lemma}

Since $S(z, w)= \overline{S(w, z)}$,  Lemma \ref{lem-rep-hardy} implies
\begin{align}\label{rem-S-self}
\int_\D S(w, z) |K_\D(z, w)|^2 dV(w) = S(z, z) K_\D(z,z).
\end{align}

\begin{proof}[Proof of Lemma \ref{lem-rep-hardy}]
Set
\[
I_1(z) := \int_\D S(z, w) |K_\D(z, w)|^2 dV(w).
\]
The rotational invariance of the Lebesgue measure gives
\begin{align}\label{av-I-5}
I_1(z) = \int_\D  \underbrace{\left[\frac{1}{2 \pi} \int_0^{2 \pi} S(z, w e^{- i \theta}) | K_\D(z, w e^{- i \theta})|^2 d\theta \right]}_{\text{denoted by $I_2(z, w)$}}dV(w).
\end{align}
Now we compute $I_2(z,w)$ by using the residue method:
\begin{align*}
I_2(z, w) & = \frac{1}{2 \pi} \int_0^{2\pi} \frac{ d\theta}{ \pi^2(1 - z \bar{w} e^{i \theta})^{3} ( 1 - \bar{z} w e^{- i \theta})^2}
\\
& = \frac{1}{2 \pi i} \int_0^{2 \pi} \frac{e^{i \theta} \cdot i e^{i\theta} d\theta}{\pi^2(1 - z \bar{w} e^{i \theta})^{3} ( e^{i \theta} - \bar{z} w)^2}
\\
& = \frac{1}{2 \pi i} \oint_C \frac{H(\zeta) d\theta}{( \zeta - \bar{z}w)^2},
\end{align*}
where $C$ is the unit circle oriented counterclockwise  and $H$ is a holomorphic function defined on a neighbourhood of $\overline{\D}$  by  the formula
\[
H(\zeta)= \frac{ \zeta}{\pi^2(1 - z \bar{w}  \zeta)^{3} }.
\]
It follows that
\[
I_2(z, w) = H'(\bar{z}w) =  \frac{1}{\pi^2} \left[ \frac{1}{(1 - |zw|^2)^{3}} + \frac{ 3 | z w|^2}{(1 - | zw|^2)^{4}}\right].
\]
Therefore, we have
\begin{align*}
I_1(z) &= \int_\D  \frac{1}{\pi^2} \left[ \frac{1}{(1 - |zw|^2)^{3}} + \frac{3 | z w|^2}{(1 - | zw|^2)^{4}}\right] dV(w)
\\
& =  \frac{ 2 \pi }{ \pi^2}\int_0^1  \left[ \frac{1}{(1 - |z|^2 r^2)^{3}} + \frac{3 | z |^2 r^2}{(1 - | z|^2 r^2)^{4}}\right]  rdr
\\
& = \frac{1 }{ \pi}\int_0^1  \left[ \frac{1}{(1 - |z|^2 t)^{3}} + \frac{3 | z |^2 t}{(1 - | z|^2 t)^{4}}\right]  dt
\\
& =\frac{1}{\pi} \int_0^1 \frac{d}{dt}\left[ \frac{t}{(1 - |z|^2 t)^{3}}\right]dt
\\
&=  \frac{1}{\pi} \frac{1}{(1 - |z|^2)^{3}} = S (z, z) K_\D(z,z).
\end{align*}
\end{proof}

{\flushleft\it{Proof of Proposition \ref{prop-hardy}.}}
Set $L_S(z, w): = S(z, w) |K_\D(z, w)|^2$.  The reproducing property gives $\langle S_x, S_y  \rangle_{H^2(\D)}  = S_x(y) =  S(y, x)$.
From \eqref{f-var} and the identity
\[
K_\D(z, z) = \int_\D | K_\D(z, w)|^2 dV(w),
\] we obtain
\begin{multline*}
 \Var_{\Pi_{K_\D}} \Big[  \sum_{x\in  \X} g(x) S_x   \Big]
=  \int_\D   \|g(z)S_z\|_{H^2(\D)}^2 K_\D(z,z) dV(z)
\\
 -  \frac{1}{2}\int_\D \int_\D \langle g(z) S_z, g(w) S_w \rangle_{H^2(\D)} |K_\D(z,w)|^2 dV(z) dV(w)
\\
 - \frac{1}{2}\int_\D \int_\D \langle g(w) S_w, g(z) S_z \rangle_{H^2(\D)} |K_\D(z,w)|^2 dV(z) dV(w)
\\
=  \int_\D   g(z)^2 S(z, z) K_\D(z,z) dV(z)   -  \frac{1}{2}\int_\D \int_\D g(z)  g(w)  S(w,z)|K_\D(z,w)|^2 dV(z) dV(w)
\\
 - \frac{1}{2}\int_\D \int_\D g(w) g(z) S(z,w) |K_\D(z,w)|^2 dV(z) dV(w)
\\
=   \int_\D   g(z)^2  S(z,z) K_\D(z,z) dV(z)   -  \int_\D \int_\D g(z) g(w)   L_S(z, w) dV(z) dV(w).
\end{multline*}
Now using Lemma \ref{lem-rep-hardy} and the equality \eqref{rem-S-self}, we have
\begin{align*}
\int_\D g(z)^2 S(z,z) K_\D(z,z) dV(z) & = \int_\D \int_\D g(z)^2  L_S(z, w) dV(z) dV(w)
\\
& =\int_\D \int_\D g(w)^2  L_S(z, w) dV(z) dV(w).
\end{align*}
Consequently,
\begin{multline*}
\Var_{\Pi_{K_\D}} \Big[  \sum_{x\in  \X} g(x) S_x   \Big]  = \int_\D \int_\D  g(z)^2 L(z,w) dV(z)dV(w)
 -  \int_\D \int_\D g(z) g(w)   L_S(z, w) dV(z) dV(w)
\\
 =  \frac{1}{2}\int_\D \int_\D  g(z)^2 L_S(z,w) dV(z)dV(w)
 + \frac{1}{2}\int_\D \int_\D  g(w)^2 L_S(z,w) dV(z)dV(w)
\\
  -  \int_\D \int_\D g(z) g(w)   L_S(z, w) dV(z) dV(w)
\\
= \frac{1}{2}\int_\D\int_\D | g(z)- g(w) |^2 L_S(z,w) dV(z) dV(w). \qed
\end{multline*}

Nonnegativity of the right hand side of  \eqref{var-hardy} is not directly clear. If the function $g$ is complex-valued, then we have
\[
\Var_{\Pi_{K_\D}} \Big[  \sum_{x\in  \X} g(x) S_x   \Big]    =  \int_\D   |g(z)|^2 K_\D(z,z)^2 dV(z)   -  \int_\D \int_\D g(z) \overline{g(w)}   L(z, w) dV(z) dV(w).
\]
Since in general, we have
\[
 \int_\D \int_\D g(z) \overline{g(w)}   L(z, w) dV(z) dV(w) \ne \int_\D \int_\D \overline{ g(z)} g(w)   L(z, w) dV(z) dV(w),
\]
the equality \eqref{var-hardy} does not hold for complex-valued functions $g$.

We proceed to the proof of Proposition \ref{prop-up-var-weight}.  Given a compactly supported {radial} weight function $W:\D \rightarrow \R^{+}$, introduce  the family $(W^{z_o})_{z_o\in\D}$ of weight functions by the formula
\begin{align}\label{family-weight}
W^{z_o}(z) :=  W(\varphi_{z_o}(z)) =  W\left( \frac{z_o-z}{1  - \bar{z}_o z}\right), \quad z \in \D.
\end{align}
Note that  $W^0 = W$ and
\begin{align}\label{conf-equiv}
W^{\varphi(z_o)} (\varphi(z)) = W^{z_o}(z), \quad \varphi \in \Aut(\D).
\end{align}

{\flushleft \it{Proof of Proposition \ref{prop-up-var-weight}.}}
The Bergman kernel on the disk is conformally invariant:
\begin{align}\label{conf-inv-K}
\frac{1}{(1 - z \bar{w})^2} =  \frac{\varphi'(z) \overline{\varphi'(w)}}{( 1 - \varphi(z) \overline{\varphi(w)})^2}, \quad \varphi \in \Aut(\D).
\end{align}
Taking the square root of both sides yields
\begin{align}\label{conf-equiv-hardy}
S(z,w)  = \varphi'(z)^{\frac{1}{2}} S(\varphi(z), \varphi(w)) \overline{ \varphi'(w)}^{\frac{1}{2}}, \quad \varphi \in \Aut(\D).
\end{align}
The involution $\varphi_{z_o}(z) = (z_o - z)(1 - \bar{z_o} z)^{-1}$ satisfies
$\varphi_{z_o}'(z)= (-1 + |z_o|^2) (1 - \bar{z}_o z)^{-2}$. Set $z = \varphi_{z_o}(\eta), w = \varphi_{z_o}(\xi)$.
Proposition  \ref{prop-hardy}, the conformal equivariance  \eqref{conf-equiv} for $W^{z_o}(z)$  and the conformal invariance  \eqref{conf-inv-K} together imply
\begin{align}\label{general-var-hardy}
\begin{split}
 & \Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W^{z_o} (x) S_x  \Big]
\\
=&   \frac{1}{2} \int_\D\int_\D   |W(\eta) - W(\xi) |^2\frac{(1 - \bar{z}_o \eta ) (1 - z_o \bar{\xi})}{1- |z_o|^2} \underbrace{ S(\eta, \xi) |    K_\D( \eta, \xi)|^2}_{\text{denoted by $L_S(\eta, \xi)$}} dV(\eta) dV(\xi).
\end{split}
\end{align}
By our assumption, $W$ is radial, therefore
\[
\Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W^{z_o} (x) S_x  \Big] = \frac{1}{2(1- |z_o|^2)}\int_\D \int_\D | W(\eta) - W(\xi)|^2 I_3(\eta, \xi) dV(\eta) dV(\xi),
\]
where
\[
I_3(\eta, \xi) := \int_0^{2 \pi} \int_0^{2\pi} (1 - \bar{z}_o \eta e^{i \theta_1}) (1 - z_o \bar{\xi} e^{i \theta_2}) L_S(\eta e^{i \theta_1}, \xi e^{-i \theta_2}) \frac{d\theta_1}{2\pi} \frac{d \theta_2}{2 \pi}.
\]
We start by computing
\[
I_4(\eta, \xi) :=  \int_0^{2\pi} (1 - \bar{z}_o \eta e^{i \theta}) L_S(\eta e^{i \theta}, \xi ) \frac{d\theta}{2\pi}.
\]
We have
\begin{multline*}
I_4(\eta, \xi) = \frac{1}{2 \pi}\int_0^{2\pi} \frac{1 - \bar{z}_o \eta e^{i \theta} }{\pi^2 (1 - \eta \bar{\xi} e^{i \theta})^{3}  ( 1 - \bar{\eta} \xi e^{- i \theta})^2} d\theta
\\
 = \frac{1}{2 \pi i }\int_0^{2\pi} \frac{(1 - \bar{z}_o \eta e^{i \theta})   e^{i \theta}\cdot i e^{i \theta}  d\theta}{\pi^2 (1 - \eta \bar{\xi} e^{i \theta})^{3}  ( e^{i \theta} - \bar{\eta} \xi )^2}
= \frac{1}{2 \pi i } \oint_C \frac{  H_1(\zeta) d\zeta}{ ( \zeta - \bar{\eta} \xi)^2},
\end{multline*}
where $C$ is the unit circle oriented counterclockwise  and $H_1$ is a holomorphic function on a neighbourhood of the closed unit disk $\overline{\D}$ defined by
the formula
\[
H_1(\zeta) : = \frac{(1 - \bar{z}_o \eta \zeta)  \zeta}{\pi^2 (1 - \eta \bar{\xi} \zeta)^{3} }.
\]
It follows that
\begin{align*}
 I_4 (\eta, \xi) = H_1'(\bar{\eta}\xi)
 =  & \frac{1}{\pi^2}\left[\frac{1 - 2\bar{z}_o \eta \zeta}{ (1 - \eta \bar{\xi} \zeta)^{3} } + \frac{3 (1 - \bar{z}_o \eta \zeta)  \zeta  \eta \bar{\xi}}{ (1 - \eta \bar{\xi} \zeta)^{4} }\right]\Big|_{\zeta = \bar{\eta} \xi}
\\
=& \frac{1}{\pi^2}\left[\frac{1 - 2\bar{z}_o |\eta |^2 \xi  }{ (1 - |\eta \xi |^2)^{3} } + \frac{3 (1 - \bar{z}_o |\eta |^2 \xi)  |\eta \xi|^2}{ (1 - |\eta \xi|^2)^4 }\right].
\end{align*}
Substituting $I_4(\eta, \xi)$ into the formula for $I_3(\eta, \xi)$, we obtain
\begin{align*}
I_3(\eta, \xi)  = \frac{1}{2\pi}\int_0^{2 \pi}  (1 - z_o \bar{\xi} e^{i \theta}) I_3(\eta, \xi e^{- i \theta}) d\theta.
\end{align*}
Using the elementary identity
$$
 \frac{1}{2\pi}\int_0^{2 \pi}  ( 1- z_o \bar{\xi} e^{i \theta}) \cdot  ( 1 - \bar{z}_o | \eta|^2 \xi e^{- i \theta})  d\theta = 1 + |z_o \eta\xi|^2,
$$
for any $\eta, \xi\in \D$, we obtain
\[
 I_3(\eta, \xi) =  \frac{1}{\pi^2} \left[  \frac{1    +   2 | z_o \eta \xi|^2}{ (1 - |\eta \xi |^2)^3 } + \frac{3 |\eta \xi|^2 (1 + |z_o \eta \xi|^2)}{ (1 - |\eta \xi|^2)^{4} } \right]  =  \frac{1 + | z_o \eta \xi|^2 ( 2- | \eta \xi|^2)}{\pi^2 (1 - |\eta \xi|^2)^4}. \qed
\]

\begin{proof}[Proof of Corollary \ref{prop-decay-var-hardy}]
By Proposition \ref{prop-up-var-weight}, we have
\[
\Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W_s^{z_o} (x) S_x  \Big]   \le \frac{3}{2 \pi^2(1- |z_o|^2)}\int_\D \int_\D  \frac{| W_s(\eta) - W_s(\xi)|^2 }{ (1 - |\eta \xi|^2)^4} dV(\eta) dV(\xi)
\]
We have
\begin{align*}
& \int_\D \int_\D    \frac{| W_s(\eta) - W_s(\xi)|^2}{  ( 1 - |\eta \xi|^2)^{4}} dV(\eta) dV(\xi)
\\
  = &  4 \pi^2 \int_0^1 \int_0^1 \frac{ \Big| ( 1 - x^2)^s \mathds{1}(x^2 \le 2 - s) - ( 1 - y^2)^s \mathds{1}(y^2 \le 2 - s) \Big |^2 }{( 1 - x^2 y^2)^{4}} xy dxdy
\\
= & \pi^2  \int_0^1 \int_0^1 \frac{ \Big| ( 1 - x)^s \mathds{1}(x \le 2 - s) - ( 1 - y)^s \mathds{1}(y \le 2 - s) \Big |^2 }{( 1 - x y)^{4}} dxdy
\\
= & \pi^2 \underbrace{ \int_0^1 \int_0^1 \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{( x + y -x y)^{4}} dxdy}_{\text{denoted by $I_5$}},
\end{align*}
where we have  changed variables twice, first $(x, y)\to (x^2, y^2)$, then $(x, y)\to (1 - x, 1-y)$.
The integrand in $I_5$ is symmetric in the variables $x$ and $y$, and we have
\[
I_5 = 2  \iint_{0 \le y \le x \le 1} \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{( x + y -x y)^{4}}  dxdy.
\]
Using the elementary inequality $x  + y - xy \ge x$, we obtain
\begin{align*}
I_5 \le &  2 \iint_{0 \le y \le x \le 1} \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{x^{4}}  dxdy
\\
=&   2 \underbrace{ \iint_{s-1 \le y \le x \le 1} \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{x^{4}}  dxdy}_{I_{6}}
\\
& + 2 \underbrace{\iint_{ 0 \le y \le s-1 \le x \le 1} \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{x^{4}}  dxdy}_{I_{7}}
\\
& + 2  \underbrace{\iint_{ 0 \le y \le x \le  s-1 } \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{x^{4}}  dxdy}_{=0}.
\end{align*}
Applying the change of variables $x = x', y = \lambda x'$ to the integral $I_{6}$, and noting that $\det \frac{\partial( x, y) }{\partial (x', \lambda)} = x'$ , we obtain that
\begin{align*}
I_{6}  = \int_{s-1}^1 dx \int_0^1 d\lambda  [  x^{2s-3} ( 1 - \lambda^s)^2] \le \frac{1- (s-1)^{2s-2}}{2s -2}   \le \frac{1 - (s-1)^{s-1}}{s-1}.
\end{align*}
We estimate $I_{7}$ under the assumption $1 < s < \frac{5}{4}$. We have
\begin{align*}
I_{7} & = \iint_{ 0 \le y \le s-1 \le x \le 1} \frac{ \Big|  x^s \mathds{1}(x \ge  s-1) - y^s \mathds{1}(y \ge  s-1) \Big |^2 }{x^{4}}  dxdy
\\
& = \iint_{ 0 \le y \le s-1 \le x \le 1}   x^{2s-4}   dxdy= (s-1) \frac{x^{2s-3}}{2s - 3}\Big|_{x=s-1}^1 \le \frac{(s-1)^{2s-2}}{3 - 2s} \le \frac{1}{3-2s}.
\end{align*}
By the conformal invariance of the distribution of $\Pi_{K_\D}$,  we have
\begin{align*}
\E_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W_s^{z_o} (x)  \Big]     &  =  \E \Big[    \sum_{x\in  \X} W_s (x)  \Big]   = \int_\D ( 1 - |z|^2)^s \mathds{1}(|z|^2 \le 2 - s) K_\D(z, z) dV(z)
\\
&  =  \int_{s-1}^1 x^{s-2}dx = \frac{ 1- (s-1)^{s-1}}{s -1} .
\end{align*}
Therefore, we have
\[
\frac{ \displaystyle \Var_{\Pi_{K_\D}} \Big[    \sum_{x\in  \X} W_s^{z_o} (x) S_x  \Big]  }{\displaystyle \Big[  \E_{\Pi_{K_\D}} \Big(    \sum_{x\in  \X} W^{z_o} (x)   \Big)   \Big]^2} \le    \frac{3}{2 \pi^2(1- |z_o|^2)} \frac{2 \pi^2 I_{6} +  2 \pi^2 I_{7}}{\Big[  \frac{ 1- (s-1)^{s-1}}{s -1} \Big]^2} \le  \frac{3\pi^2}{(1- |z_o|^2)} \frac{ \frac{1 - (s-1)^{s-1}}{s-1} +  \frac{1}{3-2s}}{\Big[  \frac{ 1- (s-1)^{s-1}}{s -1} \Big]^2}.
\]
If $s\in (1, \frac{5}{4})$, then $\frac{1}{3-2s} \le 2$. The equality
$
\lim_{s\to 1^+} \frac{1 - (s-1)^{s-1}}{-(s-1)\log(s-1)} =1
$
yields \eqref{log-2-decay}.
\end{proof}

\subsection{Impossibility of simultaneous reconstruction of Bergman functions}\label{sec-failure}
Proposition \ref{prop-failure} shows the impossibility of  simultaneous reconstruction of  all  $A^2(\D)$-functions  by  averaging with compactly supported radial weights.
Recall the notation \eqref{family-weight}.
\begin{proposition}\label{prop-failure}
For any $z_o\in \D$, we have
\[
\inf_{W} \frac{\displaystyle \Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big]   }{  \displaystyle  \Big[  \E_{\Pi_{K_\D}} \Big( \sum_{x\in \X} W^{z_o} (x)  \Big) \Big]^2 } > 0,
\]
where the infimum is over the  all compactly supported bounded radial weights $W: \D\rightarrow \R^{+}$.
\end{proposition}

\subsubsection{Variance of $A^2(\D)$-vector-valued linear statistics}

\begin{lemma}\label{lem-rep}
For any $z\in\D$, the Bergman kernel $K_\D$ satisfies
\[
\int_{\D} K_\D(z, w) | K_\D(z, w)|^2 dm_{2} (w) = K_\D(z, z)^2.
\]
\end{lemma}

{\flushleft \it{Proof}.}
Since the Lebesgue measure is rotationally invariant, we have
\begin{align}\label{id-sym}
\begin{split}
I_8(z): &  = \int_\D K_\D(z, w) | K_\D(z, w)|^2 dV(w)
\\
&  = \frac{1}{2\pi}\int_0^{2 \pi}  \int_\D    K_\D(z, w e^{- i \theta}) | K(z, w e^{- i \theta})|^2 dV(w) d\theta
= \int_\D I_9 (z, w) dV(w),
\end{split}
\end{align}
where
\[
 I_9(z, w): =  \frac{1}{2\pi}\int_0^{2 \pi} K_\D(z, w e^{- i \theta}) | K(z, w e^{- i \theta})|^2  d\theta.
\]
Write
\begin{multline*}
I_9(z,w)=   \frac{1}{2 \pi } \int_0^{2 \pi} \frac{1}{\pi^3} \frac{ d \theta}{( 1 - z \bar{w} e^{i \theta})^4   ( 1 - \bar{z} w e^{- i \theta})^2}
=\\= \frac{1}{2   \pi i   } \int_0^{2 \pi} \frac{1}{\pi^3} \frac{  e^{i \theta} \cdot i  e^{i \theta}d \theta}{( 1 - z \bar{w} e^{i \theta})^4  (   e^{i \theta} - \bar{z} w )^2}
= \frac{1}{2   \pi i   } \oint_{C} \frac{1}{\pi^3} \frac{  \zeta d \zeta}{( 1 - z \bar{w}  \zeta)^4  (   \zeta - \bar{z} w )^2},
\end{multline*}
where $C$ is the unit circle oriented counterclockwise. Setting  $F_{z\bar{w}} (\zeta): = \pi^{-3}  \zeta (1 - z \bar{w} \zeta)^{-4}$ and  computing the residue, we obtain
\begin{multline}\label{id-res}
I_9(z,w)  =  \frac{1}{2   \pi i   } \oint_{C}  \frac{   F_{z \bar{w}}(\zeta) d \zeta}{ (   \zeta - \bar{z} w )^2}
=   \frac{1}{2   \pi i   } \oint_{C}  \frac{   F_{z \bar{w}}(\zeta) - F_{z \bar{w}}(\bar{z} w)  -   F_{z \bar{w}}'(\bar{z} w) ( \zeta - \bar{z} w) }
{ (   \zeta - \bar{z} w )^2}   d\zeta +
\\
 + \frac{1}{2   \pi i   } \oint_{C}  \frac{    F_{z \bar{w}}(\bar{z} w)  +   F_{z \bar{w}}'(\bar{z} w) ( \zeta - \bar{z} w) }{ (   \zeta - \bar{z} w )^2}   d\zeta
= F_{z \bar{w}}' ( \bar{z} w) = \frac{1}{\pi^3} \left[  \frac{1}{(1 - | z w|^2)^4} + \frac{4 |zw|^2}{ ( 1 - |zw|^2)^5} \right] .
\end{multline}
Substituting \eqref{id-res} into \eqref{id-sym}, we rewrite $I_8(z)$ in the form
\begin{multline*}
  \int_{\D} \frac{1}{\pi^3} \left[  \frac{1}{(1 - | z w|^2)^4} + \frac{4 |zw|^2}{ ( 1 - |zw|^2)^5} \right] dV(w)
 = \frac{2\pi}{\pi^3} \int_0^1 \left( \frac{1}{(1 - | z|^2  r^2 )^4} + \frac{4 |z|^2 r^2 }{ ( 1 - |z|^2 r^2)^5} \right) rdr =
\\
 = \frac{1}{\pi^2} \int_0^1 \left( \frac{1}{(1 - | z|^2  t )^4} + \frac{4 |z|^2 t }{ ( 1 - |z|^2 t)^5} \right) dt
= \frac{1}{\pi^2} \int_0^1  \frac{d}{dt} \left( \frac{t}{(1 - | z|^2  t )^4} \right) dt  =\\= \frac{1}{\pi^2} \frac{1}{(1 - |z|^2)^4} = K_\D(z, z)^2.\qed
\end{multline*}

Let  $K_\D^x \in A^2(\D)$ be defined by $K_\D^x ( z) = K_\D(z, x)$.  The reproducing property gives
\begin{align}\label{rep-property}
\langle K_\D^x, K_\D^y  \rangle_{A^2(\D)}  = K_\D^x(y) =  K_\D(y, x).
\end{align}
To a compactly supported bounded Borel function $g: \D\to \mathbb C$ assign the $A^2(\D)$-vector-valued linear statistics
\begin{equation}\label{vec-add}
\sum_{x\in \X} g(x) K_\D^x.
\end{equation}
If $g$ takes real values, then by the same argument used in the proof of Proposition \ref{prop-hardy}, we obtain  the variance of the linear statistic \eqref{vec-add} in the following
\begin{proposition}\label{prop-Sob}
If $g: \D \rightarrow \R$ is a compactly supported bounded Borel function, then
\begin{align}\label{var-rep-stat}
\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} g(x) K_\D^x  \Big] = \frac{1}{2} \int_\D \int_\D | g(z) - g(w)|^2 \cdot   K_\D(z, w) |K_\D(z, w)|^2 dV(z) dV(w).
\end{align}
\end{proposition}

\subsubsection{Estimates of variance}

We would like to estimate
\[
\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big].
\]
Recall that the  M{\"o}bius involution $\varphi_{z_o}\in \Aut(\D)$,  $\varphi_{z_o}(z) = (z_o - z)(1 - \bar{z_o} z)^{-1}$
satisfies $\varphi_{z_o}'(z)= (-1 + |z_o|^2) (1 - \bar{z}_o z)^{-2}$. By Proposition \ref{prop-Sob} and  \eqref{conf-inv-K}, \eqref{conf-equiv},
writing $z = \varphi_{z_o}(\eta)$, $w = \varphi_{z_o}(\xi)$  gives, for the variance
$\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big]$,  the expression
 \begin{multline}\label{general-var}
 \frac{1}{2} \int_\D\int_\D   |W^{z_o}(z) - W^{z_o}(w) |^2 K_\D(z, w) | K_\D(z, w)|^2 dV(z) dV(w)=
\\
=   \frac{1}{2} \int_\D\int_\D   |W^{0}(\eta) - W^{0}(\xi) |^2\frac{(1 - \bar{z}_o \eta )^2 (1 - z_o \bar{\xi})^2}{(1- |z_o|^2)^2} K_\D(\eta, \xi) |    K_\D( \eta, \xi)|^2 dV(\eta) dV(\xi).
\end{multline}

As in \eqref{var-rep-stat}, positivity of \eqref{general-var} is not quite immediate;  see Lemma \ref{lem-res-2} below.
Rotational invariance of the function $W^0$  gives for $\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big]$ the expression
\begin{align}\label{var-non-sym}
 \frac{1}{2 ( 1 - |z_o|^2)^2} \int_\D \int_\D | W^0(\eta) - W^0(\xi)|^2  \cdot I_{10}(\eta, \xi) dV(\eta) dV(\xi),
\end{align}
where $I_{10}(\eta, \xi)$ is given by
\[
I_{10}(\eta, \xi) = \int_0^{2 \pi} \int_0^{2 \pi} (1 - \bar{z}_o \eta  e^{i \theta_1})^2 (1 - z_o \bar{\xi}  e^{-i \theta_2}  )^2 K_\D(\eta e^{i \theta_1}, \xi e^{i \theta_2})  | K_\D( \eta e^{i \theta_1}, \xi e^{i \theta_2})|^2   \frac{d \theta_1}{2 \pi} \frac{d \theta_2}{2 \pi}.
\]

\begin{lemma}\label{lem-res-1}
For any $\eta, \xi \in \D$, set
\[
I_{11}(\eta, \xi): = \frac{1}{2\pi} \int_0^{2 \pi}  (1 - \bar{z}_o \eta  e^{i \theta})^2  K_\D(\eta e^{i \theta}, \xi )  | K_\D( \eta e^{i \theta}, \xi )|^2   d \theta.
\]
Then we have
\begin{align}\label{I-4-id}
I_{11}( \eta, \xi) = \frac{1}{\pi^3} \left[  \frac{(1 - \bar{z}_o | \eta|^2 \xi) ( 1 - 3 \bar{z}_o |\eta|^2 \xi)}{( 1 - |\eta \xi|^2)^4}  + \frac{4 |\eta \xi|^2 ( 1 - \bar{z}_o | \eta|^2 \xi)^2}{( 1 - |\eta \xi|^2)^5}\right].
\end{align}
\end{lemma}
{\flushleft\it{Proof}.}
Write
\begin{multline*}
I_{11}(\eta, \xi)  = \frac{1}{2 \pi}\int_0^{2\pi}  \frac{1}{\pi^3} \frac{(1 - \bar{z}_o \eta e^{i \theta})^2}{ ( 1 - \bar{\xi}  \eta e^{i \theta})^4 ( 1 - \xi \bar{\eta} e^{- i \theta})^2  } d\theta =
\\
  = \frac{1}{2 \pi i }\int_0^{2\pi}  \frac{1}{\pi^3} \frac{(1 - \bar{z}_o \eta e^{i \theta})^2  e^{ i \theta}\cdot i  e^{ i \theta}
   d\theta  }{ ( 1 - \bar{\xi}  \eta e^{i \theta})^4 ( e^{i \theta} - \xi \bar{\eta} )^2  }
 = \frac{1}{2 \pi i }\oint_C \frac{1}{\pi^3} \frac{(1 - \bar{z}_o \eta  \zeta)^2  \zeta   d\zeta  }{ ( 1 - \bar{\xi}  \eta \zeta)^4 ( \zeta - \xi \bar{\eta} )^2  },
\end{multline*}
where $C$ the unit circle oriented counterclockwise.  Setting
$$G (\eta): = \frac{1}{\pi^3} \frac{(1 - \bar{z}_o \eta  \zeta)^2  \zeta    }{ ( 1 - \bar{\xi}  \eta \zeta)^4  },$$
 we obtain for $I_{11}(\eta, \xi)$ the expression
$$
\frac{1}{2 \pi i }\oint_C  \frac{ G(\zeta)   }{( \zeta - \xi \bar{\eta} )^2  } d \zeta
= G'(\xi \bar{\eta}) = \frac{1}{\pi^3} \left[  \frac{(1 - \bar{z}_o | \eta|^2 \xi) ( 1 - 3 \bar{z}_o |\eta|^2 \xi)}{( 1 - |\eta \xi|^2)^4}  + \frac{4 |\eta \xi|^2 ( 1 - \bar{z}_o | \eta|^2 \xi)^2}{( 1 - |\eta \xi|^2)^5}\right].\qed
$$

\begin{lemma}\label{lem-res-2}
For any $\eta, \xi \in \D$, we have
\begin{align}\label{I-3-id}
I_{10}(\eta, \xi) = \frac{1}{\pi^3} \left[ \frac{1 + 8 |z_o \eta \xi|^2 + 3 |z_o \eta \xi|^4}{ ( 1 - | \eta \xi|^2)^4} + \frac{4 | \eta \xi|^2 ( 1 + 4 |z_o \eta \xi|^2 + | z_o \eta \xi|^4)}{(1 - |\eta \xi|^2)^5}\right].
\end{align}
\end{lemma}

\begin{proof}
By definition of $I_4(\eta, \xi)$, we have
\begin{equation}\label{i10-expr}
I_{10}(\eta, \xi) = \frac{1}{2 \pi} \int_0^{2 \pi}  ( 1 - z_o \bar{\xi} e^{- i \theta})^2 I_4( \eta, \xi e^{i \theta}) d\theta.
\end{equation}
Substitute \eqref{I-4-id} into \eqref{i10-expr}.  The orthonormality of $(e^{i n \theta})_{n\in\Z}$ in $L^2( [0, 2 \pi], \frac{d\theta}{2 \pi})$ gives
\[
I_{10}(\eta, \xi) =  \frac{1}{\pi^3} \left[ \frac{1 + 8 |z_o \eta \xi|^2 + 3 |z_o \eta \xi|^4}{ ( 1 - | \eta \xi|^2)^4} + \frac{4 | \eta \xi|^2 ( 1 + 4 |z_o \eta \xi|^2 + | z_o \eta \xi|^4)}{(1 - |\eta \xi|^2)^5}\right].
\]
\end{proof}
From
\eqref{var-non-sym} we directly obtain
\begin{proposition}\label{prop-new-var-f}
For any  $z_o\in \D$, we have
\[
\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big] =\frac{1}{2 ( 1 - |z_o|^2)^2} \int_\D \int_\D | W(\eta) - W(\xi)|^2  \cdot I_{10}(\eta, \xi) dV(\eta) dV(\xi),
\]
where $I_{10}(\eta, \xi)$ is given by the formula \eqref{I-3-id}.
\end{proposition}
\begin{corollary}\label{var-est-vec}
Therefore, we have the following estimates:
\begin{align}\label{var-ineq}
\Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big] \approx  \frac{1}{( 1 - |z_o|^2)^2} \int_\D \int_\D \frac{|W(\eta) - W(\xi)|^2}{( 1 - | \eta \xi|^2)^5} dV(\eta) dV(\xi),
\end{align}
where $\approx$ means that there exist $0 < c< C<\infty$ such that the ratio of the left and right hand sides of \eqref{var-ineq} is in the interval $[c, C]$.
\end{corollary}
{\flushleft \it{Proof of Corollary \ref{var-est-vec}.}} We have
\[
I_{10}(\eta, \xi) \le \frac{1}{\pi^3} \frac{36}{(1 - | \eta \xi|^2)^5}
\]
Setting $t = |z_o \eta \xi|^2 \in [0,1)$ gives
\begin{multline*}
I_{10}(\eta, \xi) \ge \frac{1}{\pi^3 ( 1 - | \eta \xi|^2)^5} \left[ ( 1 - | \eta \xi|^2) (1 + 8 t + 3 t^2 ) + 4 | \eta \xi|^2 ( 1 + 4 t + t^2) \right]\geq \\
\\ \ge \frac{1}{\pi^3 ( 1 - | \eta \xi|^2)^5}\left[ 1  + 8 t + 3 t^2 + |\eta \xi|^2 ( 3 + 8t + t^2)\right]  \ge \frac{1}{\pi^3 ( 1 - | \eta \xi|^2)^5}.\qed
\end{multline*}

\subsubsection{Proof of Proposition \ref{prop-failure}}
Set  $V(y): = W(\sqrt{1-y})$ and $g(y)  = \frac{V(y)}{y^2}$ for $y \in (0, 1)$. Since $W$ is compactly supported,
there exists $\varepsilon> 0$ such that  $\supp (g) \subset [\varepsilon, 1]$. We have
\begin{align*}
 \E_{\Pi_{K_\D}} \Big( \sum_{x\in \X} W^{z_o} (x)  \Big) &  =  \E_{\Pi_{K_\D}} \Big( \sum_{x\in \X} W (x)  \Big) = \int_\D \frac{W(z)}{ \pi ( 1 - |z|^2)^2} dV(z)  = 2 \int_0^1 \frac{W(r)}{(1-r^2)^2} rdr
\\
&  =\int_0^1 \frac{W(\sqrt{x})}{(1-x)^2}dx =  \int_0^1 \frac{W(\sqrt{1-y})}{ y^2} dy = \int_0^1 \frac{V(y)}{y^2}dy  = \int_0^1 g(y) dy.
\end{align*}
and
\begin{align*}
H_W: & = \int_\D \int_\D \frac{|W(\eta) - W(\xi)|^2}{( 1 - | \eta \xi|^2)^5} dV(\eta) dV(\xi) = 4 \pi^2 \int_0^1 \int_0^1 \frac{ |W(r_1) - W(r_2)|^2}{(1 - r_1^2 r_2^2)^5}  r_1  r_2 dr_1 dr_2
\\
&  = \pi^2 \int_0^1 \int_0^1 \frac{ |W(\sqrt{x}) - W(\sqrt{y})|^2}{(1 - xy)^5}  dx dy.
\end{align*}
Using change of variables $x = 1 - x', y = 1 - y'$, we obtain
\begin{align*}
H_W & = \pi^2 \int_0^1 \int_0^1 \frac{ |V(x) - V(y)|^2}{( x + y  - xy)^5}  dx dy = 2 \pi^2 \int_{0 \le x \le y \le 1} \frac{ |V(x) - V(y)|^2}{( x + y  - xy)^5}  dx dy
\\
&  \ge 2 \pi^2 \int_{0 \le x \le y \le 1} \frac{ |V(x) - V(y)|^2}{( x + y  )^5}  dx dy.
\end{align*}
Now using change of variables $x = \lambda y, y = y$ for the last integral, we obtain
\begin{align*}
H_W   & \ge 2\pi^2 \int_0^1 \int_0^1 \frac{ | V(\lambda y) - V(y)|^2}{ (\lambda + 1)^5 \cdot y^4} d\lambda dy  \ge \frac{\pi^2}{16} \int_0^1 \int_0^1 \frac{ | V(\lambda y) - V(y)|^2}{  y^4} d\lambda dy
\\
&  = \frac{\pi^2}{16}\int_0^1\int_0^1 \left|  \lambda^2  g(\lambda y) - g(y) \right|^2 d\lambda dy.
\end{align*}
Note that
\begin{align*}
\int_0^1\int_0^1 \left|  \lambda^2  g(\lambda y)  \right|^2 d\lambda dy = \int_0^1 \lambda^3 d\lambda \int_0^\lambda g(x)^2 dx \le \frac{1}{4} \int_0^1 g(x)^2 dx.
\end{align*}
Therefore, by the triangle inequality, we have
\begin{align*}
& \left(\int_0^1\int_0^1 \left|  \lambda^2  g(\lambda y) - g(y) \right|^2 d\lambda dy\right)^{1/2}
\\
 \ge&   \left(\int_0^1\int_0^1 \left|  g(y) \right|^2 d\lambda dy\right)^{1/2} - \left(\int_0^1\int_0^1 \left|  \lambda^2  g(\lambda y)\right|^2 d\lambda dy\right)^{1/2}
\\
 \ge &  \frac{1}{2} \left(\int_0^1 g(y)^2 dy\right)^{1/2} \ge \frac{1}{2} \int_0^1 g(y) dy.
\end{align*}
Consequently, we  have
\begin{align*}
 \frac{\displaystyle \Var_{\Pi_{K_\D}} \Big[ \sum_{x\in \X} W^{z_o} (x)  K_\D^x  \Big]   }{  \displaystyle  \Big[  \E_{\Pi_{K_\D}} \Big( \sum_{x\in \X} W^{z_o} (x)  \Big) \Big]^2 }  \ge  \frac{\pi^2}{16} \cdot \frac{1}{4}  = \frac{\pi^2}{64}. \qed
\end{align*}

\section{Patterson-Sullivan measures}\label{sec-PS}
\subsection{Weak convergence}
 Let $(M, d)$ be a proper  complete metric space and let $o \in M$ be a fixed point.   We say that a point process $\Pi$ on $M$ is  {\it asymptotically  controlled by $\mu_M$} if the first correlation function $\rho_1^{(\Pi)}(x)$ with respect to the reference measure $\mu_M$ converges uniformly to a positive constant as $d(x, o) \to \infty$.  For any Radon measure $\frak{m}$ on $M$,  let $\Stab(\frak{m})$ denote the subgroup of $\Iso(M)$ consisting of all isometries of $M$ preserving the Radon measure $\frak{m}$.   The main result of this section is
\begin{theorem}\label{thm-general}
 Let $(M, d)$ be a proper  complete metric space. Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and Assumption \ref{ass-M-A2}.  Let $\Pi$ be a point process on $M$ asymptotically controlled by $\mu_M$ and satisfying Assumption \ref{ass-pp}. Then $\Pi$-almost every configuration $X$ on $M$ is of divergent type with critical exponent $\delta_\PS(X)  =  h_M$.    Moreover,  for any fixed sequence $\mathfrak{s}=(s_n)_{n\in\N}$ such that
\[
\sum_{n=1}^\infty (s_n- h_M)^2 < \infty,
\]
there exists a subset $\Omega_{\mathfrak{s}} \subset \Conf(M)$ satisfying $\Pi(\Omega_{\mathfrak{s}}) = 1$  such that for any $X \in \Omega_{\mathfrak{s}}$,  any $y \in M$, we have
\[
\lim_{n\to\infty}    \frac{ \displaystyle \sum_{x\in X} e^{-s_n d(y, x)}  \delta_x }{ \displaystyle \sum_{x\in X} e^{- s_n d(o, x)}}   =  e^{- h_M B_\xi (y, o)} \mu_o (d\xi).
\]
 In particular, we have the following weak convergence of probability measures:
\[
\lim_{n\to\infty}    \frac{ \displaystyle \sum_{x\in X} e^{-s_n d(y, x)}  \delta_x }{ \displaystyle \sum_{x\in X} e^{- s_n d(y, x)}} =  \frac{\displaystyle   e^{- h_M B_\xi (y, o)} \mu_o (d\xi)}{\displaystyle  \int_{\partial M} e^{- h_M B_\xi (y, o)} \mu_o (d\xi)}.
\]
\end{theorem}

\begin{remark}\label{lem-conf-density}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and Assumption \ref{ass-M-A2}.  Then  the map defined by \eqref{def-conf-density} is  an $\Stab(\mu_M)$-conformal density  of dimension $h_M$.
\end{remark}

\subsection{Identification of the limit measures}\label{sec-lim-measure}

\begin{lemma}\label{lem-ac-density}
Let $\mu_M$ be a Radon measure on $M$ satisfying Assumption \ref{ass-M-A1}. Let   $\sigma$ be any  accumulation point, as $s \to h_M^{+}$,  of the family of probability measures
\begin{align}\label{family-exp-m}
\left(  \frac{ e^{-s d(x, o)} \mu_M(dx)}{ \displaystyle \int_M e^{-s d(x, o)} \mu_M(dx)}   \right)_{ s \in (h_M, \infty)}.
\end{align}
Then the family $(\widehat{\mu}_y)_{y\in M}$ defined by
\begin{align}\label{def-intro-density}
\widehat{\mu}_y (d\xi) = e^{-h_M B_\xi (y, o) } \sigma (d\xi)
\end{align}
is a normalized $\Stab(\mu_M)$-conformal density  of dimension $h_M$.
\end{lemma}

\subsubsection{Critical exponent}
For a positive Radon measure $\frak{m}$ on $M$, we define its critical exponent by
\[
\delta_\PS(\frak{m}): = \inf\Big\{s > 0\Big| \int_M e^{-s d(x, y)} \frak{m}(dy)< \infty \Big\} \in [0, \infty],
\]
with the convention $\inf \emptyset = + \infty$. The  critical exponent $\delta_\PS(\frak{m})$ does not depend on the choice of $x\in M$.
We say that $\frak{m}$ is of divergent type if
\[
\delta_\PS(\frak{m})<\infty \an \int_M e^{- \delta_\PS (\frak{m}) \cdot d(x, y)} \frak{m}(dy)  = \infty,
\]
for some and hence all $x\in M$.

\begin{remark}
The assumption \eqref{ass-rate}  immediately implies that
\begin{align}\label{exp-and-div}
\delta_\PS(\mu_M) = h_M  \an \text{$\mu_M$ is of divergent type.}
\end{align}
\end{remark}

For a closed subgroup $G< \Iso(M)$, we define its critical exponent by
\[
\delta_\PS(G):  = \inf\Big\{s> 0\Big| \int_G e^{-s d(x, gy)} dg<\infty\Big\} \in [0, \infty],
\]
where $dg$ denotes the Haar measure of $G$, unique up to a multiplicative constant. The definition of $\delta_\PS(G)$ does not depend on the choices of $x, y \in M$.   We say that $G$ is of divergent type if
\[
\delta_\PS(G)<\infty \an \int_G e^{- \delta_\PS (G) \cdot d(x, gy)} dg   = \infty
\]
for some and hence all $(x, y) \in M^2$.
The critical exponent of a Radon measure and that of a closed subgroup of $\Iso(M)$ are related as follows. Assume that $G< \Stab(\frak{m})$ is a closed unimodular subgroup of $\Stab(\frak{m})$. Consider the Haar measure $dg$ on $G$ and let  $[y]$ be the image of $y$ under the quotient map $M \to M/G$.  By \cite[Lemma 6.1]{Burger-Mozes} there exists  a unique Radon measure  $\widehat{\frak{m}}$ on the quotient space $ M/G$  such that
\begin{align}\label{m-decom}
\int_M \varphi (y) \frak{m}(dy)=  \int_{M/G} \widehat{\frak{m}} (d [y]) \int_G  \varphi( g y) dg, \quad \text{for all $\varphi \in L^1(M, \frak{m})$}.
\end{align}
In particular, if $G$ acts cocompactly on $M$, then $\widehat{\frak{m}}$ is a finite measure.
\begin{proposition}\label{prop-m-stab}
Let $\frak{m}$ be  a positive Radon measure on $M$. Assume that there exists  a closed unimodular subgroup  $G < \Stab(\frak{m})$ which acts cocompactly on $M$.  Then $\delta_\PS(\frak{m}) = \delta_\PS(G)$ and  $\frak{m}$ is of divergent type if and only if $G$ is of divergent type.
\end{proposition}
\begin{proof}
The decomposition \eqref{m-decom} implies
\begin{equation}\label{cmp-cnv-m}
\int_M e^{-s d (x, y)} \frak{m}(dy) = \int_{M/G} \widehat{\frak{m}} (d[y]) \int_G e^{-s d(x, gy)} dg.
\end{equation}
Fix a point $x \in M$.  If,  for some $s> 0$, the left hand side of \eqref{cmp-cnv-m} converges, then there exists $y_0 \in M$ such that $\int_G e^{-s d(x, gy_0)} dg <\infty$, whence  $\int_G e^{-s d(x, gy)} dg<\infty$ for all $y \in M$.  Now assume that there exists $y_0 \in M$ such that $\int_G e^{-s d(x, gy_0)} dg < \infty$. Then $\int_G e^{-s d(x, gy)} dg< \infty$ for all $y\in M$ and moreover the function $y \mapsto \int_G e^{-s d(x, gy)} dg$ is continuous. Compactness of $M/G$ implies  that the right hand side of \eqref{cmp-cnv-m} converges, and, consequently,  so does the left hand side.
\end{proof}

\subsubsection{Analysis of Assumption \ref{ass-M-A1}}

 We say that the measure $\mu_M$ has purely exponential  growth with respect to the base point $o \in M$, if \eqref{ass-rate} is satisfied for the pair of constant functions $(L, U) = (c, C)$ with $0< c < C<\infty$. The  following Propostion is a direct consequence of  Roblin \cite[Th\'eor\`eme 4.1.1]{Roblin}.

\begin{proposition}\label{prop-pure-exp}
Let $M$ be a proper geodesic $\catone$ space and  let $\mu_M$ be a positive Radon measure on $M$. If a discrete subgroup $\Gamma < \Stab(\mu_M)$ acts cocompactly on $M$ then   $\mu_M$ has purely exponential growth with respect to the base point $o \in M$.
\end{proposition}
\begin{proof}
Since $\Gamma<\Stab(\mu_M)$ acts cocompactly on $M$, it has a finite Bowen-Margulis measure $m_\Gamma$ and by Proposition \ref{prop-m-stab} and \eqref{exp-and-div}, we have $\delta_\PS(\Gamma) = h_M$. Therefore, by  \cite[Corollaire 2 of Th\'eor\`eme 4.1.1]{Roblin}, we have
\begin{align}\label{exp-number}
 h_M \| m_\Gamma\| e^{-h_M t}  \#\Big\{\gamma \in \Gamma \Big| d(o, \gamma y) \le t\Big\} \xrightarrow{t \to\infty}  \| \mu_y\|.
\end{align}
Since $\Gamma$ acts cocompactly on $M$, there exists a relatively compact Borel fundamental domain $F\subset M$. By integrating \eqref{exp-number} with respect to $\mu_M|_F$, using  the equalities $\gamma_{*} (\mu_M |_F) = \mu_M|_{\gamma(F)}$ for all $\gamma \in \Gamma$ and  the dominated convergence theorem, we obtain
\begin{align}\label{integrating-number}
\begin{split}
 & h_M \| m_\Gamma\| e^{- h_M t} \int_F  \#\Big\{\gamma \in \Gamma \Big| d(o, \gamma y) \le t\Big\}   \mu_M(dy)
\\
&  =  h_M \| m_\Gamma\| e^{- h_M t}  \mu_M(B(o, t))  \xrightarrow{t \to\infty}  \int_F \| \mu_y\| \mu_M(dy).
\end{split}
\end{align}
It follows that $\mu_M(B(o, t))$ has purely exponential growth rate.
\end{proof}

The following theorem of  Knieper gives further  examples satisfying Assumption \ref{ass-M-A1}. Recall that an Hadamard manifold is by definition a simply connected manifold
of nonpositive curvature.
\begin{theorem}[{Knieper \cite[Theorem A]{Knieper}}]
Let $M$ be a nonflat Hadamard manifold admitting a discrete
subgroup $\Gamma<\Iso(M)$ such that $M/\Gamma$ is a compact smooth manifold. Then for any $x \in M$, there exists $a > 1$ and $r_0 >0$ such that
\begin{align}\label{exp-sphere}
  a^{-1} r^{\frac{ \rank(M)-1}{2}}    e^{h r} \le  s(r; x) \le  a  r^{\frac{ \rank(M)-1}{2}}   e^{h r}, \quad \text{for $r \ge r_0,$}
\end{align}
where $s(r; x)$ denotes the volume of the geodesic sphere centred at $x$ with radius $r$, $h > 0$ denotes the volume entropy of $M$ and $\rank(M)$ denotes the rank of the manifold $M$.
\end{theorem}
The growth rate \eqref{exp-sphere} of the volume of the geodesic spheres clearly implies the growth rate of volume of the geodesic balls: for any $x\in M$, there exists $C>1$ and $r_0 >0$, such that
\[
C^{-1} r^{\frac{ \rank(M)-1}{2}}    e^{h r} \le  b(r; x) \le  C r^{\frac{ \rank(M)-1}{2}}   e^{h r}, \quad \text{for $r \ge r_0,$}
\]
where $b(r; x)$ denotes the volume of the geodesic ball centred at $x$ with radius $r$.

\subsubsection{Analysis of Assumption \ref{ass-M-A2}.}
 Since $\overline{M}$ is compact, the family \eqref{family-exp-m} is tight in $\mathfrak{M}^1(\overline{M})$. The assumption \eqref{ass-rate} now also implies that, under the weak topology  on the space of probability measures on $\overline M$, any accumulation point for the family \eqref{family-exp-m} as $s$ approaches to $h_M$ from above, is  supported on $\partial M$.  Nevertheless, this accumulation point  might in general not be unique. Under Assumption \ref{ass-M-A2}, if
\[
\lim_{r \to\infty }  \frac{ \mathds{1}_{B(o, r)} \mu_M(dx)}{\mu_M( B(o, r))}  = \sigma_o,
\]
then, cf. Proposition \ref{prop-unif-exp} in the Appendix, we have the weak convergence \eqref{def-S-o} to the limit $\mu_o= \sigma_o$.

For a  proper geodesic $\catone$ space $M$, Roblin \cite[Corollaire 1.8]{Roblin} proved that any discrete subgroup $\Gamma < \Iso(M)$ of divergent type has a unique normalized $\Gamma$-conformal density of dimension $\delta_\PS(\Gamma)$, while Burger and Mozes proved
\begin{theorem}[{Burger and Mozes \cite[Theorem 6.2 and Theorem 6.4]{Burger-Mozes}}]\label{BM-result}
If $M$ is a proper geodesic $\catone$ space, then any closed unimodular nonelementary subgroup $G < \Iso(M)$ of divergent type has a unique normalized $G$-conformal density of dimension $\delta_\PS(G)$.
\end{theorem}

Theorem \ref{prop-pure-exp}, Theorem \ref{BM-result},  Remark \ref{lem-conf-density}  and  \eqref{exp-and-div} together imply

\begin{proposition}\label{prop-exp-cv}
If $M$ is a proper geodesic $\catone$ space and $\mu_M$ be a positive Radon measure admitting closed discrete non-elementary
subgroup $G< \Stab(\mu_M)$ acting cocompactly on $M$, then
 $\mu_M$ has purely exponential growth rate, the weak convergence \eqref{def-S-o} holds, and the map \eqref{def-conf-density} defines the unique normalized $\Stab(\mu_M)$-conformal density of dimension $h_M$.
\end{proposition}

\subsection{Proof of Lemma \ref{lem-ac-density}}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and Assumption \ref{ass-M-A2}. Let $\sigma$ be any accumulation point of the family \eqref{family-exp-m} as $s$ approaches to $h_M$ from above.   Fix a sequence $(s_n)_{n\ge 1}$ in $(h_M, \infty)$ such that $\lim_{n\to\infty} s_n = h_M$ and the following weak convergence of probability measures holds:
\begin{align}\label{weak-ap-bis}
\lim_{n\to\infty }\frac{ \displaystyle e^{-s_n d(x, o)} \mu_M(dx)}{ \displaystyle \int_M e^{-s_n d(x, o)} \mu_M( d x)}   = \sigma(d\xi).
\end{align}
We return to the family  $(\widehat{\mu}_y)_{y\in M}$ of regular Borel measures given by the formula
\[
\widehat{\mu}_x (d \xi) = e^{- h_M B_\xi(x, o)} \sigma(d\xi),
\]
where  $B_\xi(x, y)$ is the Busemann cocycle. For any $x, y \in M$, we have
\[
\frac{d\widehat{\mu}_y}{d\widehat{\mu}_x} (\xi)  = e^{-h_M B_\xi(y, x)}.
\]
The family $(\widehat{\mu}_y)_{y\in M}$ is normalized with respect to the base point $o \in M$ since $\widehat{\mu}_o = \sigma$ is by definition a probability measure. Now to finish the proof of Lemma \ref{lem-ac-density}, we need to show that the family $(\widehat{\mu}_y)_{y\in M}$ is $\Stab(\mu_M)$-equivariant.

{\flushleft \bf Claim I:} {\it We have
\begin{align}\label{claim-1}
\widehat{\mu}_y (d\xi) = \lim_{n\to\infty} \frac{  \displaystyle e^{-s_n d(x, y)} \mu_M(dx)}{ \displaystyle \int_M e^{-s_n d(x, o)} \mu_M(d x)}.
\end{align}
}
Let $\Psi: \overline{M} \rightarrow \R$  the function defined by
\begin{align}\label{cont-Psi}
\Psi (x) =     d(x, y) - d(x, o)  \text{\, if $x\in M$}\an \Psi(\xi ) = B_\xi (y, o) \text{\, if $\xi \in \partial M$}.
\end{align}
Note that $\Psi \in C(\overline{M})$. Take any $f \in C(\overline{M})$, we have
\begin{align}\label{one-to-hm}
\begin{split}
& \limsup_{n\to\infty} \left|  \frac{  \displaystyle \int_M e^{-s_n d(x, y )}  f(x) \mu_M(d x )}{\displaystyle{\int_M e^{-s_n d(x, o)} \mu_M(dx)} }  -   \frac{  \displaystyle \int_M e^{-h_M\Psi(x)}  f(x)  e^{-s_n d(x, o)} \mu_M(d x )}{\displaystyle{\int_M e^{-s d(x, o)} \mu_M(dx)} }  \right|
\\
& \le   \| f\|_{C(\overline{M})}  \limsup_{n\to\infty}   \|  e^{-s_n \Psi  } - e^{- h_M \Psi } \|_{C(\overline{M})}.
\end{split}
\end{align}
Using  the inequality $| e^t -1|\le e^{|t|}-1$ for all $t\in \R$, we have
\begin{align*}
  \|  e^{-s \Psi  } - e^{- h_M \Psi } \|_{C(\overline{M})}  \le \|   e^{- h_M \Psi } \|_{C(\overline{M})} \Big( e^{(s - h_M) \|\Psi\|_{C(\overline{M})}}  - 1\Big),
\end{align*}
and thus
\begin{align}\label{exp-to-zero}
\limsup_{n\to\infty} \|  e^{-s_n \Psi  } - e^{- h_M \Psi } \|_{C(\overline{M})}  = 0.
\end{align}
Now since $e^{-h_M \Psi} \cdot f \in C(\overline{M})$, by \eqref{weak-ap-bis},  \eqref{one-to-hm} and \eqref{exp-to-zero}, we have
\begin{align*}
\lim_{n\to\infty}   \frac{  \displaystyle \int_M e^{-s_n d(x, y )}  f(x) \mu_M(dx )}{\displaystyle{\int_M e^{-s_n d(x, o)} \mu_M(dx)} }   & = \lim_{n\to\infty}    \frac{  \displaystyle \int_M e^{-h_M\Psi(x)}  f(x)  e^{-s_n d(x, o)} \mu_M( d x )}{\displaystyle{\int_M e^{-s_n d(x, o)} \mu_M(dx)} }
\\
&   =   \displaystyle {\int_{\partial M}  f ( \xi ) e^{- h_M \beta_\xi(y)}   \sigma(d \xi)} = \int_{\partial M} f(\xi) \widehat{\mu}_y (d\xi).
\end{align*}

{\flushleft \bf Claim II:} {\it For any $\gamma \in \Stab(\mu_M)$ and any $y\in M$, we have $\gamma_{*}\widehat{\mu}_y = \widehat{\mu}_{\gamma(y)}$. }

Indeed, by using  twice \eqref{claim-1},  the continuity of the action of $\gamma$ on $\overline{M}$,  the $\Stab(\mu_M)$-invariance of $\mu_M$ and the equality $d(x, y) = d(\gamma(x), \gamma(y))$ for all $x, y \in M$, we have
\begin{align*}
\gamma_{*}\widehat{\mu}_y (d\xi) = \lim_{n\to\infty}  \gamma_{*} \left( \frac{ \displaystyle e^{-s_n d(x, y)} \mu_M(dx)}{ \displaystyle \int_M e^{-s_n d(x, o)} \mu_M(d x)}\right) = \lim_{n\to\infty}   \frac{  \displaystyle e^{-s_n d(x, \gamma(y))} \mu_M(dx) }{ \displaystyle \int_M e^{-s_n d(x, o)} \mu_M(d x)} = \widehat{\mu}_{\gamma(y)}(d\xi). \qed
\end{align*}
\subsection{Proof of Theorem \ref{thm-general}}\label{sec-proof-PS}
\subsubsection{Variance of linear statistics}
\begin{proposition}\label{prop-sup-var}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and $\Pi$ be a point process on $M$ asymptotically controlled by $\mu_M$ and satisfying Assumption \ref{ass-pp}.  Then for any $f \in C(\overline{M})$ and any $y \in M$, we have
\begin{align}\label{var-up}
\sup_{s \in (h_M, \infty)} \Var_{\Pi}\Big(\sum_{x \in \X} e^{- s d(y, x)} f(x) \Big) < \infty.
\end{align}
\end{proposition}
\begin{lemma}\label{lem-mean-order}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and $\Pi$ be a point process on $M$ asymptotically controlled by $\mu_M$. Then for any $y \in M$, the inequality $\E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(y, x)}\Big] <\infty$ holds if and only if $s >h_M$.
 Moreover,
\begin{align}\label{lim-inf-linear-st}
\liminf_{s \to h_M^{+}}  \left( (s - h_M) \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(y, x)}\Big] \right) > 0.
\end{align}
\end{lemma}

\begin{proof}
Recall that there exist $h_M >0$ and a controllable pair $(L, U)$ of continuous positive functions,
 such that $L(r) e^{r \cdot h_M } \le \mu_M(B(o, r)) \le   U(r) e^{r\cdot h_M}$ for all $r>0$.
Since $\liminf_{r\to\infty }L(r) > 0$, there exist $R_0>0, \delta > 0$ such that $L(r) \ge \delta$ for any $r \ge R_0$.   Since $\Pi$ is asymptotically controlled by $\mu_M$, there exist $\rho>0, R\ge R_0$  such that
\begin{align}\label{asy-conf-inv}
 \rho \le \rho_1^{(\Pi)}(x) \le 2 \rho, \quad \text{for all $x \in B(o, R)^c$.}
\end{align}
Since $e^{-s d(o, y)} \cdot e^{-s d(o, x)} \le e^{-s d(y, x)} \le e^{s d(o, y)} \cdot e^{-s d (o, x)}$, we have
\[
e^{-s d(o, y)} \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big] \le \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(y, x)}\Big] \le   e^{s d(o, y)}  \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big].
\]
The bound \eqref{asy-conf-inv} and  the identity \eqref{int-rep-exp}  imply, for any $s> h_M$, the inequalities
\begin{multline*}
\E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(y, x)}\Big]   \le   e^{s d(o, y)}  \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big]\le
\\
\le e^{s d(o, y)}  \left[   \int_{B(o, R)} \rho_1^{(\Pi)}(x)  \mu_M(dx)  + \int_{B(o, R)^c} e^{-s d(o, x)} \rho_1^{(\Pi)}(x) \mu_M(dx) \right] \le
\\
\le    e^{s d(o, y)}   \int_{B(o, R)} \rho_1^{(\Pi)}(x)  \mu_M(dx)
 +    2 \rho e^{s d(o, y)}   \int_{B(o, R)^c} \Big(\int_{\R^{+}}  s e^{- s r} \mathds{1} ( d(o, x) \le r) dr \Big)\mu_M(dx) =
\\
 =   e^{s d(o, y)}   \int_{B(o, R)} \rho_1^{(\Pi)}(x)  \mu_M(dx)
 +  2 \rho e^{s d(o, y)}\int_{R}^\infty s e^{-s r} [\mu_M( B(o, r)) - \mu_M(B(o, R))] dr=
\\
=   e^{s d(o, y)}   \int_{B(o, R)} \rho_1^{(\Pi)}(x)  \mu_M(dx)  +  2 \rho e^{s  d(o, y)}\int_{R}^\infty s e^{-(s -h_M)r}   U(r)   dr < \infty;
\end{multline*}
\begin{align*}
\E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(y, x)}\Big] &  \ge   e^{-s d(o, y)}  \E_{\Pi}\Big[\sum_{x \in \X}  e^{-s d(o, x)}\Big]
\\
&
\ge e^{-s d(o, y)}  \int_{B(o, R)^c} e^{-s d(o, x)} \rho_1^{(\Pi)}(x) \mu_M(dx)
\\
& \ge      \rho e^{-s d(o, y)}   \int_{B(o, R)^c} \Big(\int_{\R^{+}}  s e^{- s r} \mathds{1} ( d(o, x) \le r) dr \Big)\mu_M(dx)
\\
& \ge \rho e^{-s d(o, y)}\int_{R}^\infty s e^{-s r} [\mu_M( B(o, r)) - \mu_M(B(o, R))] dr
\\
& \ge   \rho \delta e^{-s d(o, y)} \int_{R}^\infty s e^{-(s-h_M) r} dr -   \rho   \mu_M(B(o, R)) e^{-s d(o, y) -s R}
\\
& =  \frac{\rho \delta s   e^{-s d(o, y) - sR +h_M R}}{s-h_M}  -  \rho   \mu_M(B(o, R)) e^{-s d(o, y) -s R},
\end{align*}
 and \eqref{lim-inf-linear-st} follows.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-sup-var}]
Since $\Pi$ satisfies Assumption \ref{ass-pp}, there exists $C>0$, such that for any $f \in C(\overline{M})$, we have
\[
\Var_{\Pi}\Big(\sum_{x \in \X} e^{- s d(y, x)} f(x) \Big) \le C  \| f\|_{C(\overline{M})}^2\int\limits_M e^{-2s d(y, x)}  \rho_1^{(\Pi)}(x) \mu_M(dx).
\]
It follows that
\[
\sup_{s \in (h_M, \infty) } \Var_{\Pi}\Big(\sum_{x \in \X} e^{- s d(y, x)} f(x) \Big) \le  C  \| f\|_{C(\overline{M})}^2\int\limits_M e^{-2 h_M d(y, x)}  \rho_1^{(\Pi)}(x) \mu_M(dx).
\]
Since $\Pi$ is asymptotically controlled by $\mu_M$, by definition, we have
\[
\int_M e^{-2 h_M d(y, x)} \rho_1^{(\Pi)}(x)  \mu_M(dx) \le  e^{ 2 h_M d(o, y)}\int_M e^{-2 h_M d(o, x)} \rho_1^{(\Pi)}(x) \mu_M(dx) <\infty,
\]
where in the last inequality for the finiteness of the integral, we used Lemma \ref{lem-mean-order}.  The proof of the proposition is complete.
\end{proof}

\subsubsection{Configurations of divergent type}

\begin{proposition}\label{prop-div}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1}.  Assume that $\Pi$ is a point process asymptotically controlled by $\mu_M$ and satisfying Assumption \ref{ass-pp}. Then  $\Pi$-almost every configuration is of divergent type with  critical exponent $h_M$.
\end{proposition}

Let $\Pi$ be a point process on $M$. We say that counting function of $\Pi$ satisfies the law of large numbers if for any sequence  $(I_k)_{k\ge 1}$ of relatively compact measurable subsets of $M$ such that
\begin{align}\label{large-mean}
\lim_{k\to\infty}  \E_\Pi(\#(I_k \cap \X)) = \infty,
\end{align}
we have
\[
\lim_{k\to\infty}  \left\|  \frac{\#(I_k \cap \X)}{\E_\Pi(\#(I_k \cap \X))} -1\right\|_{2}^2 =  \lim_{k\to\infty}\frac{\Var_\Pi(\#(I_k \cap \X))}{[\E_\Pi(\#(I_k \cap \X))]^2} = 0.
\]
In what follows we will only use  the convergence of the counting function in probability:  for any $\varepsilon>0$,
\[
\lim_{k\to\infty} \Pi \left(\Big| \frac{\#(I_k \cap \X)}{\E_\Pi(\#(I_k \cap \X))} -1\Big| \ge \varepsilon \right) =0.
\]

Note that if $\Pi$ is a point process on $M$ satisfying Assumption \ref{ass-pp}, then the counting function of $\Pi$ satisfies the law of large numbers. Indeed, for any relatively compact measurable subset $I \subset M$,  by taking $f  = \chi_I$ in  the inequality \eqref{var-less-2-moment}, we have
\begin{align}\label{var-mean}
\Var_\Pi\Big(\#(I \cap \X)\Big) \le C \E_\Pi\Big(\#(I\cap \X)\Big).
\end{align}
The law of large numbers for the counting function of $\Pi$ then follows immediately.

\begin{proof}[Proof of Proposition \ref{prop-div}]
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and $\Pi$ be a point process on $M$ asymptotically controlled by $\mu_M$.  By Lemma \ref{lem-mean-order}, for $\Pi$-almost every $X$,
\[
\sum_{x \in X}  e^{-s d(o, x)}<\infty, \quad \text{for all $s > h_M$.}
\]
Therefore, $\delta_{\PS}(X) \le h_M$ for $\Pi$-almost every $X$. It remains to show that
\[
\sum_{x\in X} e^{- h_M d(o, x)} = \infty, \quad \text{for $\Pi$-almost every $X$.}
\]
Now by Assumption \ref{ass-M-A1}, we may choose a large enough $r_0 > R_0$ such that there exists $\delta>0$ and we have $L(k r_0) \ge \delta > 0$ and $e^{r_0 h_M} L( (k+1)r_0)  \ge 2 U(k r_0)$ for any $k \in \N$  and such that \eqref{asy-conf-inv} holds for any $x \in B(o, r_0)^c$.  Set $A_k: = B(o, (k+1) r_0) \setminus B(o, k r_0)$.
Then for any  positive integer $k$, we have
\begin{multline*}
\E_\Pi ( \# ( \X \cap A_k))     = \int_{A_k}  \rho_1^{(\Pi)} (x) d\mu_M(x)    \ge   \frac{\rho}{2} \mu_M(A_k)
  = \mu_M ( B(o, (k+1) r_0)) - \mu_M ( B (o, k r_0))
\\
  \ge  (e^{r_0 h_M} L( (k+1)r_0)  - U(k r_0)) e^{k r_0 h_M}
\ge  U(kr_0)  e^{k r_0 h_M} \ge  L(kr_0)  e^{k r_0 h_M} \ge \delta e^{k r_0 h_M}.
\end{multline*}
Hence $\E_\Pi ( \#_{A_k}) \to \infty$ as $k\to\infty$.  By \eqref{var-mean}, the counting function of $\Pi$ satisfies the law of large numbers and thus there exists $\Omega \subset \Conf(M)$ with $\Pi(\Omega) = 1$ and  a subsequence $(A_{k_\ell})_{\ell \ge 1}$, such that
\[
\frac{\#(A_{k_\ell} \cap X)}{\E_\Pi (\#( A_{k_\ell} \cap \X))} \ge \frac{1}{2} \quad \text{for all $X\in \Omega$}.
\]
 Therefore,  for any $X\in \Omega$, we have
\begin{multline*}
\sum_{x\in X} e^{- h_M d(o, x)}    \ge  \sum_{\ell=1}^\infty \sum_{x\in X \cap A_{k_\ell}} e^{- h_M d(o, x)}  \ge \sum_{\ell=1}^\infty  e^{- h_M (k_\ell + 1)r_0}  \#(X \cap A_{k_\ell})
\\
 \ge \frac{1}{2} \sum_{\ell=1}^\infty  e^{- h_M (k_\ell + 1)r_0} \E_\Pi(\#(\X \cap A_{k_\ell}))
 \ge \frac{\delta }{2} e^{-h_M r_0}    \sum_{\ell=1}^\infty 1   = \infty.
\end{multline*}
The proof is complete.
\end{proof}

\subsubsection{Convergence of measures}

Our main purpose here is to obtain, for $\Pi$-almost every configuration $X$,  the weak convergence of the probability measures
\[
\left(    \Big( \sum_{x \in X} e^{-s d(y, x)} \Big)^{-1}   \sum_{x\in X} e^{-s d(y, x)} \delta_x     \right)_{s > h_M}.
\]

\begin{proposition}[Convergence of deterministic measures]\label{prop-as-inv}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying Assumption \ref{ass-M-A1} and Assumption \ref{ass-M-A2}.  Let $\Phi: M \rightarrow \R^{+}$ be a measurable function which is locally integrable with respect to the measure $\mu_M$. Assume that there exists $C> 0$ such that $\lim_{d(x, o) \to \infty} \Phi(x) = C$.
Then for any $y \in M$, we have the following weak convergence of probability measures:
\[
\lim_{s \to  h_M^{+}}\frac{  \displaystyle e^{-s d(x, y)} \Phi(x) \mu_M(dx)}{\displaystyle{\int_{M}} e^{-s d(x, y)}   \Phi(x) \mu_M(dx)} = \bar{\mu}_{y}(d \xi ).
\]
\end{proposition}

\begin{proof}
Using the same argument in the proof of  the weak convergence \eqref{claim-1}, under Assumption \ref{ass-M-A1} and Assumption \ref{ass-M-A2},  we have
\begin{align}\label{cv-exp-m}
\lim_{s \to  h_M^{+}}\frac{  e^{-s d(x, y)} \mu_M(d x)}{\displaystyle{\int_{M}} e^{-s d(x, o)}    \mu_M(dx)} = \mu_{y}(d \xi) = e^{-h_M B_\xi (y, o)} \mu_o(d\xi).
\end{align}
In particular, we have
\begin{align}\label{ratio-weight}
\lim_{s \to  h_M^{+}}\frac{  \displaystyle \int_M  e^{-s d(x, y)} \mu_M(d x)}{\displaystyle{\int_{M}} e^{-s d(x, o)}    \mu_M(dx)}  = \mu_y (\partial M).
\end{align}
Let $f \in C(\overline{M})$ be any continuous positive function  on $M$. Then for any $R>0$, we have
\begin{align*}
& \left| \frac{\displaystyle{\int_M} e^{-s d(y, x) }  f(x) \Phi(x) \mu_M(dx) }{\displaystyle{ C  \cdot \int_M} e^{-s d(y, x) }  f(x) \mu_M(dx) } - 1 \right|
\\
 \le &   \frac{  \|f\|_{C(\overline{M})}  \displaystyle{\int_{B(o, R)}}  |\Phi(x)- C| \mu_M(dx) }{\displaystyle{ C \cdot \int_M} e^{-s d(y, x) }  f(x) \mu_M(dx) } + \frac{\displaystyle{\int_{B(o, R)^c}} e^{-s d(y, x) }  f(x) |\Phi(x)- C| \mu_M(dx) }{\displaystyle{ C\cdot \int_M} e^{-s d(y, x) }  f(x)  \mu_M(dx) }
\\
\le &  \frac{  \|f\|_{C(\overline{M})}  \displaystyle{\int_{B(o, R)}}  |\Phi(x)- C| \mu_M(dx) }{\displaystyle{  C\cdot \int_M} e^{-s d(y, x) }  f(x) \mu_M(dx) }  + \frac{\displaystyle\sup_{x \in B(o, R)^c} | \Phi(x) - C|}{C}.
\end{align*}
By first letting $s\to h_M^+$ and then letting $R\to\infty$, we obtain that
\begin{align}\label{C-for-f}
\lim_{s\to h_M^+}  \frac{\displaystyle{\int_M} e^{-s d(y, x) }  f(x) \Phi(x) \mu_M(dx) }{\displaystyle{ C \cdot \int_M} e^{-s d(y, x) }  f(x) \mu_M(dx) }  = 1.
\end{align}
In particular, taking $f \equiv 1$, we have
\begin{align}\label{C-for-1}
\lim_{s\to h_M^+}  \frac{\displaystyle{\int_M} e^{-s d(y, x) }  \Phi(x) \mu_M(dx) }{\displaystyle{  C\cdot \int_M} e^{-s d(y, x) }  \mu_M(dx) }  = 1.
\end{align}
Combining the equalities  \eqref{C-for-f} and \eqref{C-for-1},  then applying \eqref{cv-exp-m} and \eqref{ratio-weight}, we obtain
\begin{align}\label{non-R-lim}
\begin{split}
& \lim_{s\to h_M^+}  \frac{\displaystyle{\int_M} e^{-s d(y, x) }  f(x) \Phi(x) \mu_M(dx) }{\displaystyle{\int_M} e^{-s d(y, x) }   \Phi(x) \mu_M(dx) }   =
\lim_{s\to h_M^+}  \frac{\displaystyle{\int_M} e^{-s d(y, x) } f(x)  \mu_M(dx) }{\displaystyle{\int_M} e^{-s d(y, x) }  \mu_M(dx) }
\\
& =  \frac{\displaystyle \int_{\partial M} f(\xi) \mu_y(d\xi)}{\mu_y(\partial M)} = \int_{\partial M} f(\xi) \bar{\mu}_y(d\xi).
\end{split}
\end{align}
The extension of the equality \eqref{non-R-lim}  from continuous positive functions to all continuous functions is direct.
\end{proof}

\begin{corollary}\label{cor-m-ration}
Let $\Pi$ be a point process on $M$ asymptotically controlled by $\mu_M$. Then for any $y\in M$, we have the following weak convergence:
\begin{align}\label{mean-Poi-Sz}
 \lim_{s\to h_M^{+}}\frac{\displaystyle \E_\Pi \Big(\sum_{x\in \X} e^{- s d (x, y)}  \delta_x \Big)}{  \displaystyle \E_\Pi \Big( \sum_{x\in \X} e^{-s  d (x, y)}\Big)}  = \bar{\mu}_y.
\end{align}
\end{corollary}

\begin{proof}
Denote by $ \rho_1^{(\Pi)}$ the first intensity function of $\Pi$  with respect to the reference measure $d\mu_M$.   Now fix $y \in M$. For any $s>h_M$,  we have
\[
\frac{ \displaystyle \E_\Pi \Big( \sum_{x\in \X} e^{- s d (y, x)}  \delta_x\Big)}{\displaystyle   \E_\Pi \Big( \sum_{x\in \X} e^{-s  d (y, x)}\Big)} =  \frac{  \displaystyle  e^{-s d (y, z)}   \rho_1^{(\Pi)}(z) d\mu_M(z)}{\displaystyle{\int_{M}}   e^{-s d (y, z)}  \rho_1^{(\Pi)}(z) d\mu_M(z)}.
\]
By Proposition \ref{prop-as-inv}, we get the desired relation \eqref{mean-Poi-Sz}.
\end{proof}

{\it {Proof of Theorem \ref{thm-general}.}}
Write $\bar{\mu}_{y}(f) = \int_{\partial M} f d \bar{\mu}_y$.
For any $s>h_M, y \in M, f \in C(\overline{M}), X\in \Conf(M)$, set
\[
\mathscr{R}(s, y, f; X) : = \frac{\displaystyle  \sum_{x \in X}  e^{-s d(y, x)}f(x)}{   \displaystyle  \sum_{x\in X} e^{- s d (y, x)} }  \an \underline{\mathscr{R}}(s, y, f; X) : = \frac{\displaystyle  \sum_{x \in X}  e^{-s d(y, x)}f(x)}{   \displaystyle  \E_\Pi\Big( \sum_{x\in \X} e^{- s d (y, x)} \Big)}
\]
Lemma \ref{lem-mean-order},  Proposition \ref{prop-sup-var}  and the assumption $\sum_k (s_k - h_M)^2 <\infty$ together imply that for any $y \in M$ and any $f\in C(\overline{M})$, we have
\[
 \sum_{k=1}^\infty  \Var_\Pi \Big( \underline{\mathscr{R}}(s_k, y, f; \X) \Big)   <\infty.
\]
Consequently, for $\Pi$-almost every $X$, we have for any $y\in M$ and $f\in C(\overline{M})$,
\begin{align}\label{Borel-cantelli-l2}
\sum_{k=1}^\infty \left|  \underline{\mathscr{R}}(s_k, y, f; X) - \E_\Pi \Big(  \underline{\mathscr{R}}(s_k, y, f; \X) \Big)\right|^2 <\infty.
\end{align}
In particular, the convergence \eqref{Borel-cantelli-l2} combined with Corollary \ref{cor-m-ration} implies that for $\Pi$-almost every $X$ and  for any $y\in M$ and $f\in C(\overline{M})$,
\begin{align}\label{half-way}
\lim_{k\to\infty}   \underline{\mathscr{R}}(s_k, y, f; X)=  \bar{\mu}_y(f).
\end{align}
Taking $f \equiv 1$ in \eqref{half-way} and combining  with \eqref{half-way} for general $f\in C(\overline{M})$, we obtain that for $\Pi$-almost every $X$, for  any $y \in M$ and any $f \in C(\overline{M})$, the limit relation
\begin{align}\label{almost-done}
\lim_{k\to\infty} \mathscr{R}(s_k, y, f; X)  =  \bar{\mu}_y(f)
\end{align}

 Let $ \mathscr{A} \subset M$ and $\mathscr{D}\subset C(\overline{M})$ be any fixed countable dense subsets.  Then, by setting
\begin{align}\label{def-Omega}
\Omega: =   \bigcap_{\stackrel{f \in \mathscr{D}}{ y \in  \mathscr{A}}}\left\{X \in \Conf(M):  \lim_{k\to\infty} \mathscr{R}(s_k, y, f; X) =  \bar{\mu}_y(f)\right\},
\end{align}
we have $\Pi(\Omega) = 1$.

We proceed to show that $\Omega$ satisfies the required property. First note that for any $y\in \mathscr{A}$, the limit relation  \eqref{almost-done} holds for all $X \in \Omega$ and all $f \in C(\overline{M})$. Indeed,  fix $y \in \mathscr{A}$ and $X \in \Omega$, then for any $f\in C(\overline{M})$, there exists a sequence $(f_\ell)_{\ell=1}^\infty$ in $\mathscr{D}$ such that $\lim_{\ell \to \infty} \| f- f_\ell\|_{C(\overline{M})} = 0$.
Then
\begin{multline*}
  \left|  \mathscr{R}(s_k, y, f; X) -  \bar{\mu}_y(f)\right|
  \le  \left|  \mathscr{R}(s_k, y, f; X)   -   \mathscr{R}(s_k, y, f_\ell; X) \right|
\\
  +  \left|  \mathscr{R}(s_k, y, f_\ell; X)  - \bar{\mu}_y (f_\ell)\right| + | \bar{\mu}_y(f_\ell) - \bar{\mu}_y(f)|
\\
\le 2 \| f - f_\ell\|_{C(\overline{M})} + \left| \mathscr{R}(s_k, y, f_\ell; X)  - \bar{\mu}_y (f_\ell)\right|.
\end{multline*}
Hence we have
\[
\limsup_{k\to\infty}  \left|   \mathscr{R}(s_k, y, f; X)  -  \bar{\mu}_y (f)\right|   \le 2 \| f- f_\ell\|_{C(\overline{M})}.
\]
Since $\ell$ is arbitrary, we obtain the desired limit relation \eqref{almost-done} for $y \in \mathscr{A}, X \in \Omega$ and any $f \in C(\overline{M})$.

Now we show that for any $X\in \Omega$ and any $y \in M$, the limit relation \eqref{almost-done} holds for any $f \in C(\overline{M})$. Writing $f = g_1 - g_2 +  \sqrt{-1} (g_3 - g_4)$ if necessary, we may assume that $f$ is non-negative. Choose a sequence $(y^{(m)})_{m=1}^\infty$ in $\mathscr{A}$ such that $\lim_{k\to\infty} d ( y^{(m)}, y) = 0$.
Then on the one hand, we have
\[
   \mathscr{R}(s_k, y, f; X)
 \le  e^{2 s_k d(y, y^{(m)})}  \mathscr{R}(s_k, y^{(m)}, f; X) .
\]
Hence for any $m\in \N$, we have
\[
\limsup_{k\to\infty}   \mathscr{R}(s_k, y, f; X)
 \le e^{2 h_M d(y, y^{(m)})} \bar{\mu}_{y^{(m)}}(f).
\]
Since $m\in\N$ is arbitrary and $\lim_{m\to\infty} \bar{\mu}_{y^{(m)}}(f) = \bar{\mu}_y (f)$, we obtain
\[
\limsup_{k\to\infty} \mathscr{R}(s_k, y, f; X)   \le \bar{\mu}_y (f).
\]
Similarly, the inequality
\[
 \mathscr{R}(s_k, y, f; X)
 \ge   e^{-2s_k d(y, y^{(m)})} \mathscr{R}(s_k, y^{(m)}, f; X)
\]
gives
\[
\liminf_{k\to\infty} \mathscr{R}(s_k, y, f; X)
 \ge \bar{\mu}_y(f),
\]
and we finally obtain the desired limit relation
\[
\lim_{k\to\infty} \mathscr{R}(s_k, y, f; X)   = \bar{\mu}_y (f). \qed
\]

\section{Appendix}\label{sec-appendix}

\subsection{Point processes}\label{sec-pp}

\subsubsection{Spaces of configurations and point processes}\label{sec-conf-pp}
Let $E$ be a locally compact metric complete separable space.
A configuration $X$ on $E$ is  a  collection of points of $E$, possibly with multiplicities and considered without regard to order,  such that any relatively compact subset  $B\subset E$ contains only finitely many points; the number of point of $X$ in $B$ is denoted $\#_B(X)$.  A configuration is called simple if all points inside have multiplicity one.  Let $\Conf(E)$ denote the space of all configurations on $E$.  A configuration $X \in \Conf(E)$  may be identified  with a purely atomic Radon measure $\sum_{x \in X} \delta_x$,  where $\delta_x$ is the Dirac mass at the point $x$, and  the space $\Conf(E)$ is a complete separable metric space with respect to the vague topology on the space of Radon measures on $E$. The Borel sigma-algebra on $\Conf(E)$ is the smallest sigma-algebra on $\Conf(E)$ that  makes all the mappings $X \mapsto \#_B(X)$ measurable, with $B$ ranging over all relatively compact Borel subsets of $E$.
A Borel probability measure $\Pi$ on $\Conf(E)$ is called a {\it point process} on $E$. A point process $\Pi$ is called simple, if $\Pi$-almost every configuration is simple.  For further background on  point processes, see, e.g.,   Daley and Vere-Jones \cite{DV-1}, Kallenberg \cite{Kallenberg}.

Let $\Pi$ be a simple point process on $E$.  For any integer $n\ge 1$, we say that a $\sigma$-finite measure $\xi_\Pi^{(n)}$ on $E^n$ is the $n$-th correlation measure of $\Pi$ if for any bounded compactly supported function $\phi: E^n \rightarrow \C$, we have
\begin{align*}
\E_\Pi\left( \sum_{x_1, \dots, x_n \in \X}^*  \phi(x_1, \dots, x_n) \right)=   \int_{E^n} \phi(y_1, \dots, y_n)  d\xi_\Pi^{(n)} (y_1, \cdots, y_n).
\end{align*}
 Endow $E$ with a reference $\sigma$-finite Radon measure $\mu$. If $\xi_\Pi^{(n)}$ is absolutely continuous to the measure $\mu^{\otimes n}$, then the Radon-Nikodym derivative
\[
\rho_n^{(\Pi)}(x_1, \cdots, x_n) := \frac{d\xi_\Pi^{(n)}}{d\mu^{\otimes n}} (x_1, \cdots, x_n) \quad \text{where $(x_1, \cdots, x_n) \in E^n$,}
\]
is called the $n$-th correlation function  of $\Pi$ with respect to the reference measure $\mu$. The first correlation measure  of a point process  is also called its first intensity.
\subsubsection{Determinantal point processes}
 Let $K$ be a {\it locally trace class positive contractive} operator on the complex Hilbert space $L^2(E,\mu)$. The local trace class assumption implies that $K$ is an integral operator and by slightly abusing the notation, we denote the kernel of the operator $K$ again by $K(x, y)$.  By  a theorem obtained by Macchi  \cite{Macchi-DP} and  Soshnikov \cite{DPP-S}, as well as by Shirai and Takahashi  \cite{ST-DPP}, the kernel $K$ induces a unique simple point process $\Pi_K$ on $E$ such that for any positive integer $n\in \N$, the $n$-th correlation function, with respect to the reference measure $\mu$,  of the point process $\Pi_K$ exists and is given by
\[
\rho_n^{(\Pi_K)} (x_1, \cdots, x_n)  = \det(K(x_i, x_j))_{1 \le i, j \le n}.
\]
The point process $\Pi_{K}$ is called  the determinantal point process on $E$ induced by the kernel $K$.

\subsubsection{Negatively correlated point processes}

\begin{proof}[Proof of Lemma \ref{lem-var-LS}]
 Let $\rho_1^{(\Pi)}$ and $\rho_2^{(\Pi)}$ be the first and second order correlation functions with respect to the reference measure $\mu$.  By the  negative correlation assumption, we have
\[
\rho_2^{(\Pi)}(x, y) \le \rho_1^{(\Pi)}(x) \rho_1^{(\Pi)}(y), \quad \text{for $(x,y) \in E \times E$.}
\]
Now let $f: E\rightarrow \C$ be such that $\E_\Pi [\sum_{x\in \X} |f(x) | + |f(x)|^2]<\infty$. Assume first that $f$ is non-negative.  We have
\begin{multline*}
 \Var_\Pi \Big(\sum_{x\in \X} f(x)\Big)  =  \int_{E} | f(x)|^2 \rho_1^{(\Pi)}(x) d\mu(x)
\\
  + \int_{E \times E} f(x) f(y) \Big[ \rho_2^{(\Pi)}(x, y) - \rho_1^{(\Pi)}(x) \rho_1^{(\Pi)}(y)\Big] \mu(dx) \mu(dy)
\\
 \le   \int_{E} | f(x)|^2 \rho_1^{(\Pi)}(x) \mu(dx) = \E_
\Pi \left( \sum_{x\in \X} | f(x)|^2 \right).
\end{multline*}
For general complex-valued function, by writing $f = g_1 - g_2 +  \sqrt{-1} (g_3 - g_4)$ with $g_1, \cdots, g_4$  non-negative such that $|f|^2  = |g_1 |^2 + | g_2|^2 + |g_3|^2 + | g_4|^2$, we obtain
\[
\Var_\Pi \Big(\sum_{x\in \X} f(x)\Big) \le 16  \E_\Pi \Big( \sum_{x\in \X} | f(x)|^2\Big).
\]
\end{proof}

\subsection{Limits of measures under  weighted averaging}
\begin{proposition}\label{prop-unif-exp}
Let $\mu_M$ be a positive Radon measure on $M$ satisfying  Assumption \eqref{ass-M-A1}. Assume that
\[
\lim_{r \to\infty }  \frac{ \mathds{1}_{B(o, r)} \mu_M(dx)}{\mu_M( B(o, r))}  = \sigma_o.
\]
Then the weak convergence \eqref{def-S-o} holds with the same limit $\mu_o= \sigma_o$.
\end{proposition}

\begin{proof}
Using the identity \eqref{int-rep-exp}, we obtain
\begin{align*}
e^{-s d(x, o)} d\mu_M(x) &=  \int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) \cdot \frac{ \mathds{1}_{B(o, r)} (x) d\mu_M(x)}{\mu_M(B(o, r))} dr
\end{align*}
and
\begin{align*}
\int_M e^{-s d(x, o)} d\mu_M(x) &=  \int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) dr.
\end{align*}
 Therefore, for any fixed $f \in C(\overline{M})$ and any $R> 0$,  we have
\begin{align}\label{exp-lim-diff}
\begin{split}
& \left | \frac{\displaystyle {\int_M f(x) e^{-s d(x, o)} d\mu_M(x)}}{\displaystyle\int_M e^{-s d(x, o)} d\mu_M(x)}   -   \int_{\partial M} f d \sigma_o  \right|
\\
\le & \frac{\displaystyle   \int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) \cdot \Big| \int_M   f(x)  \frac{ \mathds{1}_{B(o, r)} (x) d\mu_M(x)}{\mu_M(B(o, r))} - \int_{\partial M} f d \sigma_o \Big| dr }  {\displaystyle \int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) dr }
\\
\le & \frac{\displaystyle   2 \| f \|_\infty \int_{0 \le r \le R}  s e^{-s r} \mu_M(B(o, r))  dr }  {\displaystyle \int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) dr }  + \sup_{r \ge R} \Big| \int_M   f(x)  \frac{ \mathds{1}_{B(o, r)} (x) d\mu_M(x)}{\mu_M(B(o, r))} - \int_{\partial M} f d \sigma_o \Big|.
\end{split}
\end{align}
By Assumption  \ref{ass-M-A1}, we have $\int_{\R^{+}}  s e^{-s r} \mu_M(B(o, r)) dr <\infty$ if and only if $s > h_M$. By letting  $s$ tend to $h_M^{+}$ and then letting $R$ tend to infinity in the inequality \eqref{exp-lim-diff}, we obtain the desired limit relation:
\[
\lim_{s \to h_M^{+}} \frac{\displaystyle {\int_M f(x) e^{-s d(x, o)} d\mu_M(x)}}{\displaystyle\int_M e^{-s d(x, o)} d\mu_M(x)}     =  \int_{\partial M} f d \sigma_o.
\]
\end{proof}

\subsection{The mean value property for harmonic functions}

\begin{lemma}\label{lem-observation}
Let $u: \D\rightarrow \C$ be a harmonic function and $\nu$ a radial finite Radon measure on $\D$.  If $u \in L^1(\D, d\nu)$, then for any $z \in \D$ we have
\begin{align}\label{sph-mean}
u(z) = \frac{\displaystyle{\int_\D} u(\varphi_{z}(x)) d\nu(x)   }{\displaystyle{\int_\D}d\nu(x)  }.
\end{align}
In particular, if $W: \D\rightarrow \R_{+}$ is a nonzero radial  function, then for any $z\in \D$ and any harmonic function $u: \D\rightarrow \C$, we have
\begin{align}\label{mean-value-formula}
u(z) = \frac{\displaystyle{\int_\D} W(\varphi_{z}(x)) u(x)  d\mu_\D (x) }{\displaystyle{\int_\D} W(\varphi_{z}(x))d \mu_\D(x) },
\end{align}
provided  both  $W(\varphi_{z})$ and $W(\varphi_{z}) \cdot u$ are in $L^1( \D, d \mu_\D)$.
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lem-observation}]
By the radial assumption on $\nu$ and the harmonicity of the function $w \mapsto u(\varphi_{z_o} (zw))$ on a neighbourhood of $\overline{\D}$ for any fixed $z_o, z \in \D$, we have
\begin{align*}
\int_\D u(\varphi_{z_o}(z)) d\nu(z)  & = \int_0^{2 \pi} \frac{d \theta}{2 \pi} \int_\D u(\varphi_{z_o}(z e^{i \theta})) d\nu(z)  = \int_\D \left(\int_0^{2 \pi}   u(\varphi_{z_o}(z e^{i \theta}))  \frac{d \theta}{2 \pi}  \right) d\nu(z)
\\
& = u(\varphi_{z_o}(0)) \int_\D d\nu(z) = u(z_o) \int_\D d\nu(z).
\end{align*}
The equality \eqref{sph-mean} is proved.  Since $\mu_\D$ is M\"obius invariant and $\varphi_{z_o}$ is an involution M\"obius transformation on $\D$, we have
\begin{align*}
\int_\D W(\varphi_{z_o}(z)) u(z)  d\mu_\D(z)  = \int_\D W(\zeta) u(\varphi_{z_o}(\zeta))  d\mu_\D(\zeta).
\end{align*}
The equality \eqref{mean-value-formula} now follows from the equality \eqref{sph-mean} since  $W$ is radial and consequently so is the measure $W(z) \cdot d\mu_\D(z)$.
\end{proof}

\subsection{Unconditional convergence in Banach spaces}

Let $\mathscr{B}$ be a Banach space consisting of Borel functions on a standard  Borel space $E$ such that for any $f, g \in \mathscr{B}$
satisfying the pointwise inequality $|f| \le |g|$, we have $\| f \|_{\mathscr{B}} \le \| g \|_{\mathscr{B}}$.
\begin{proposition}\label{prop-positive-uncond}
Let $f_n\in \mathscr{B}$,  $f_n \ge 0$, $n\in \N$.
 If the series $\sum_{n\in \N} f_n$ converges in $\mathscr{B}$, then it converges  unconditionally.
\end{proposition}
\begin{proof}
Positivity of the terms of our series implies,  for any bijection $\tau$ of $\N$ and any $N, M \in \N$, $N\leq M$, the bound
\begin{equation}\label{perm-bnd}
\left\| \sum_{n = N}^M f_{\tau(n)} \right \|_{\mathscr{B}} \le \left \| \sum_{k = \min \{\tau (n): N \le n \le M \}}^{\max\{\tau (n): N \le n \le M \}} f_k \right\|_{\mathscr{B}}.
\end{equation}
As $N$ grows,  the minimum $\min\{ \tau(n): N \le n \le M\}$ also grows, and  (\ref{perm-bnd}) implies the convergence of
the series $\sum_{n=1}^\infty f_{\tau(n)}$  in $\mathscr{B}$. For $N\in \N$ set
\[
L_N: = \min\Big( \tau(\{1, \cdots, N\})  \Delta \{1,\cdots, N\}\Big), \quad U_N: = \max\Big( \tau(\{1, \cdots, N\})  \Delta \{1,\cdots, N\}\Big).
\]
Using positivity again, write
\begin{multline*}
\left\| \sum_{n = 1}^N  f_{\tau(n)}  - \sum_{n = 1}^N  f_{n}   \right \|_{\mathscr{B}} = \left\| \sum_{k \in \tau(\{1, \cdots, N\}) \setminus \{1,\cdots, N\}}  f_k  - \sum_{\ell \in \{1, \cdots, N\} \setminus  \tau(\{1,\cdots, N\})} f_{\ell}   \right \|_{\mathscr{B}} \le
\\
\le
 \left\| \sum_{k \in \tau(\{1, \cdots, N\}) \setminus \{1,\cdots, N\}}  f_k  \right \|_{\mathscr{B}}  +  \left\|  \sum_{\ell \in \{1, \cdots, N\} \setminus  \tau(\{1,\cdots, N\})} f_{\ell}   \right \|_{\mathscr{B}}  \le 2   \left\| \sum_{n  =  L_N}^{U_N} f_n \right \|_{\mathscr{B}}.
\end{multline*}
Since $L_N \to\infty$ as $N\to\infty$, we obtain  $\sum_{n = 1}^\infty  f_{\tau(n)}   =  \sum_{n = 1}^\infty f_{n}$.
\end{proof}

\def\cprime{$'$} \def\cydot{\leavevmode\raise.4ex\hbox{.}}