text:
\begin{document}

\date{Jun 2020}

\title{\sc Moderate Deviation estimates for\\ Nodal Lengths of\\ Random Spherical Harmonics}
\authors{\large Claudio Macci\inst{*}, Maurizia Rossi\inst{\diamond,}\footnote{Corresponding author. E-mail address: \texttt{maurizia.rossi@unimib.it}} and Anna Paola Todino\inst{\star}}

\institution{\inst{*}Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata"\\
\inst{\diamond,}Dipartimento di Matematica e Applicazioni, Universit\`a di Milano-Bicocca\\
\inst{\star}Fakult\"at f\"ur Mathematik, Ruhr-Universit\"at Bochum}

\begin{abstract}

We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics
both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020), respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Th\"ale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known
result based on the concept of exponential equivalence.

\smallskip

\noindent {\sc Keywords and Phrases:} Nodal length; Random Spherical Harmonics; Moderate Deviation Principles; Chaotic Expansions; Exponential Equivalence.

\smallskip

\noindent {\sc AMS Classification:} 60F10, 60G15, 60G60.

\end{abstract}

\section{Introduction: background and motivations}

\subsection{Random spherical harmonics}\label{sub:RSH}

Let $\mathbb S^2$ denote the two-dimensional unit sphere with the round metric and $\Delta$ the spherical Laplacian. In standard spherical coordinates $(\theta, \varphi)\in [0,\pi]\times [0,2\pi)$, where $\theta$ is the colatitude, the metric takes the form
\begin{equation}\label{metric}
dx^2 = d\theta^2 + \sin^2\theta d\varphi^2.
\end{equation}
It is well-known that the spectrum of $\Delta$ is purely discrete, its eigenvalues are of the form $-\ell(\ell+1)$ where $\ell\in \mathbb N$ and, for each $\ell$, the family of the so-called spherical harmonics of degree $\ell$ $\lbrace Y_{\ell,m}, m = 1,\dots, 2\ell +1\rbrace$ is a real orthonormal basis of the $\ell$-th eigenspace \cite[\S 3.4]{MPbook}.
\begin{definition}\label{defRSH}
For $\ell\in \mathbb N$, the $\ell$-th random spherical harmonic $T_\ell$ is a centered Gaussian field on $\mathbb S^2$ whose covariance kernel is given by
\begin{equation}\label{defCov}
\Cov(T_\ell(x), T_\ell(y)) = P_\ell(\cos d(x,y)),\quad x,y\in \mathbb S^2,
\end{equation}
where $P_\ell$ denotes the Legendre polynomial \cite[\S 13.1.2]{MPbook} of degree $\ell$ and $d(x,y)$ the spherical geodesic distance (see (\ref{metric})) between $x$ and $y$.
\end{definition}
Equivalently, one can define $T_\ell$ as follows
\begin{equation}\label{defT}
T_\ell(x) := \sqrt{\frac{4\pi}{2\ell+1} } \sum_{m=1}^{2\ell+1} a_{\ell,m} Y_{\ell,m}(x),\quad x\in \mathbb S^2,
\end{equation}
where $\{a_{\ell,m}, m=1,\dots, 2\ell+1\}$ are standard Gaussian and independent random variables, defined on a probability space $(\Omega, \mathcal F, \mathbb P)$. Actually, from the addition formula \cite[(3.42)]{MPbook} for random spherical harmonics the covariance kernel of $T_\ell$ in (\ref{defT}) is given by  (\ref{defCov}). It is immediate that $T_\ell$ is isotropic, and that it is $\mathbb P-$a.s. an eigenfunction of the spherical Laplacian with eigenvalue $-\ell(\ell+1)$. We can assume $\lbrace T_\ell, \ell \in \mathbb N\rbrace$ to be independent random fields, indeed they are the Fourier components of isotropic Gaussian fields on the sphere, see e.g. \cite{BMV07} and \cite[\S 5, \S6]{MPbook}.

Now we recall the Hilb's asymptotic formula: let $\epsilon >0$, uniformly for $\theta \in [0, \pi-\epsilon]$ and $\ell\in \mathbb N_{\ge 1}$
\begin{equation}\label{hilb}
P_\ell(\cos \theta) = \sqrt{\frac{\theta}{\sin \theta}}\, J_0 \left ( \left (\ell+1/2 \right ) \theta \right ) + O \left (\ell^{-3/2}\right ),
\end{equation}
where $J_0$ is the Bessel function \cite[\S 1.71]{Sze75} of the first kind of order zero, see the conventions below for the meaning of the $O-$notation. The scaling limit, as $\ell\to +\infty$, of $T_\ell$ is the so-called Berry's Random Wave model which, according to Berry's conjecture \cite{Ber77}, should model the local behavior of high-energy deterministic eigenfunctions on ``generic chaotic" surfaces.

\smallskip

\noindent \emph{Conventions.}  Given two sequences of positive real numbers $\lbrace x_n, n\in \mathbb N \rbrace$ and $\lbrace y_n, n\in \mathbb N\rbrace$ we will write $x_n=O(y_n)$ if the ratio $x_n/y_n$ is asymptotically bounded, and $x_n = o(y_n)$ if $\lim_{n\to +\infty} x_n/y_n=0$. Moreover, we will write $x_n \approx y_n$ if both $ x_n = O(y_n)$ and $y_n = O(x_n)$ hold.

\subsection{Nodal lengths: asymptotic distribution}\label{subsecNodal}

Let us consider the nodal set
$
T_\ell^{-1}(0) := \lbrace x\in \mathbb S^2 : T_\ell(x) = 0\rbrace.
$
It is well-known that $T_\ell^{-1}(0)$ is an $\mathbb P-$a.s. smooth curve whose connected components are homeomorphic to the circle.
We are interested in the high-energy geometry of the nodal set, in particular in the asymptotic behavior, as $\ell\to +\infty$, of the nodal length
\begin{equation}\label{nodalDef}
\mathcal L_\ell:=\text{length}(T_\ell^{-1}(0)).
\end{equation}
  The latter received great attention also in view of Yau's conjecture on nodal volumes of deterministic eigenfunctions on compact Riemannian manifolds \cite{Yau82}. We collect the main known results on the distribution of $\mathcal L_\ell$ in a single theorem (Theorem \ref{thS} below): the expected length was studied in \cite{Ber85}, the asymptotic variance in \cite{Wig10} and the second order fluctuations of $\mathcal L_\ell$ in \cite{MRW20}. In order to state them, let us recall that, for $X,Y$ integrable real random variables, the Wasserstein distance between $X$ and $Y$ is defined as $d_W(X,Y) := \sup_{h} | \mathbb E[h(X)] - \mathbb E[h(Y)] |$, where the supremum is taken over the set of Lipschitz functions whose Lipschitz constant is $\leq 1$. From now on, $Z\sim \mathcal N(0,1)$ will denote a standard Gaussian random variable.
\begin{theorem}\label{thS}
For every $\ell \in \mathbb N$
 \begin{equation}\label{mean1}
\mathbb E[\mathcal L_\ell] = \frac{4\pi}{2\sqrt 2} \sqrt{\ell(\ell+1)}.
\end{equation}
As $\ell\to +\infty$, we have
\begin{equation}\label{eqVar}
\Var(\mathcal L_\ell) =  \frac{1}{32} \log \ell \, (1+ o(1)).
\end{equation}
Moreover, denoting
 $$
\widetilde{\mathcal L}_\ell := \frac{\mathcal L_\ell - \mathbb E[\mathcal L_\ell]}{\sqrt{\Var(\mathcal L_\ell)}},
$$
a quantitative CLT in Wasserstein distance holds, i.e., as $\ell\to +\infty$
 \begin{equation}\label{Wass1}
 d_W \left ( \widetilde{\mathcal L}_\ell, Z       \right ) = O \left ( (\log \ell)^{-1/2} \right ).
 \end{equation}
\end{theorem}
Note that, from (\ref{mean1}) and (\ref{eqVar}) we have $\mathcal L_\ell/ \sqrt{\ell(\ell+1)}\to 4\pi/(2\sqrt{2})$ $\mathbb P-$a.s. as $\ell\to +\infty$, consistently with Yau's conjecture \cite{Yau82}. From (\ref{Wass1}) we conclude in particular that the asymptotic distribution of the nodal length is Gaussian. Moreover, from (\ref{Wass1}) and \cite[(C.2.6)]{NP12}
\begin{equation}
d_{Kol}\left ( \widetilde{\mathcal L}_\ell, Z       \right ) = O \left ( (\log \ell)^{-1/4} \right ),\end{equation}
where for arbitrary real random variables $X,Y$, the Kolmogorov distance between $X$ and $Y$ is defined as $d_{Kol}(X,Y) := \sup_{x\in \mathbb R} \left | \mathbb P(X\le x) - \mathbb P(Y\le x)\right |$.

\subsubsection{Shrinking spherical domains}

In \cite{Tod18} the asymptotic behavior of the nodal length in shrinking spherical domains was investigated.
Fixed a point $x_0\in \mathbb S^2$, let $B_r$ denote the spherical cap of radius $r>0$ centered at $x_0$ and consider the length of nodal lines in $B_r$
$$
\mathcal L_{\ell, B_r} := \text{length} (T_\ell^{-1}(0) \cap B_r).
$$
We recall that the area of $B_{r}$ is equal to $2\pi(1-\cos r)$. We have the following results summarized in the next theorem.
\begin{theorem}\label{thB}
For any $r>0$
\begin{equation}\label{mean2}
\mathbb E[\mathcal L_{\ell, B_r}] = \frac{2\pi(1-\cos r)}{2\sqrt{2}} \sqrt{\ell(\ell+1)}.
\end{equation}
For a sequence of radii $\lbrace r_\ell, \ell \in \mathbb N\rbrace$ converging to zero not too rapidly ($r_\ell \ell \to +\infty$) we have, as $\ell\to +\infty$,
\begin{equation}\label{var2}
\Var(\mathcal L_{\ell, B_{r_\ell}}) = \frac{1}{256} r_\ell^2 \log (r_\ell  \ell) + O (r_\ell^2).
 \end{equation}
Moreover, denoting
$$\widetilde{\mathcal L}_{\ell, B_{r_{\ell}}}:= \frac{\mathcal L_{\ell, B_{r_\ell}} - \mathbb E[\mathcal L_{\ell, B_{r_\ell}}]}{ \sqrt{\Var(\mathcal L_{\ell, B_{r_\ell}})}}$$
 we have a quantitative CLT in Wasserstein distance as $\ell\to +\infty$
\begin{equation}\label{Wass2}
d_W \left ( \widetilde{\mathcal L}_{\ell, B_{r_{\ell}}}, Z \right ) = O \left (  \left(\log r_\ell \ell \right)^{-1/2}  \right ).
\end{equation}
\end{theorem}
Indeed (\ref{Wass2}) follows from the proof of Theorem 2.2 in \cite{Tod18} (at the end of \S 5.2.3) where
in particular it is proved that the distribution of the nodal length in shrinking domains is asymptotically Gaussian. As in the previous case, we can also deduce that, as $\ell \to +\infty$,
\begin{equation}
d_{Kol}\left ( \widetilde{\mathcal L}_{\ell, B_{r_{\ell}}}, Z       \right ) = O  \left (  \left(\log r_\ell \ell \right)^{-1/4}  \right ).\end{equation}

In this paper we are interested in refinements of Central Limit Theorems stated above (Theorems \ref{thS} and \ref{thB}).

\subsection{Moderate Deviation Principles}\label{sub:MDPs}

The theory of Large Deviations allows an asymptotic computation of small
probabilities at exponential scales. Here we start by recalling a basic definition.
From now on let $(\Omega, \mathcal F, \mathbb P)$ be a complete probability space,
and $\lbrace X_n, n\in \mathbb N\rbrace$ a sequence of real-valued random variables: for each $n$ the map
$$
X_n : \Omega \to \mathbb R$$
is measurable with respect to $\mathcal F$ and $\mathcal B(\mathbb R)$ (the Borel $\sigma$-field of $\mathbb R$)
on $\Omega$ and $\mathbb R$ respectively.

\begin{definition}\label{defLDP}  We say that $\lbrace X_n, n\in \mathbb N\rbrace$ satisfies the Large Deviation Principle (LDP) with speed $0\le s_n\nearrow +\infty$ and good rate function\footnote{See \S 1.2 in \cite{DZ98} for details.} $\mathcal I$ if for every $\alpha\geq 0$ the level set $\lbrace x : \mathcal I(x) \le \alpha \rbrace$ is compact and for all $B\in \mathcal B(\mathbb R)$ we have
\begin{equation*}
- \inf_{x\in \mathring{B}} \mathcal I(x)\le \liminf_{n\to +\infty} \frac{1}{s_n} \log \mathbb P(X_n\in B) \le \limsup_{n\to +\infty} \frac{1}{s_n} \log \mathbb P(X_n\in B) \le - \inf_{x\in \bar{B}} \mathcal I(x),
\end{equation*}
where $\mathring B$ (resp. $\bar B$) denotes the interior (resp. the closure) of $B$.
\end{definition}

Let us now recall the notion of exponential equivalence \cite[Definition 4.2.10]{DZ98} related to the question whether the LDP for $\lbrace Y_n, n\in \mathbb N\rbrace$ can be deduced from the LDP for $\lbrace X_n, n\in \mathbb N\rbrace$, $\lbrace Y_n, n\in \mathbb N\rbrace$ being another sequence of real-valued random variables.
\begin{definition}\label{expdef} We say that $\lbrace X_n, n\in \mathbb N\rbrace$ and $\lbrace Y_n, n\in \mathbb N\rbrace$ are exponentially equivalent at speed $0\le s_n\nearrow +\infty$ if, for every $\delta>0$,
\begin{equation}\label{exp_equiv}
\limsup_{n\to +\infty} \frac{1}{s_n} \log \mathbb P ( | X_n - Y_n | > \delta ) = -\infty.
\end{equation}
\end{definition}
As far as the LDP is concerned, exponentially equivalent sequences of random variables are indistinguishable \cite[Theorem 4.2.13]{DZ98}.
\begin{lemma}\label{thDZ}
Assume that $\lbrace X_n, n\in \mathbb N\rbrace$ satisfies the LDP with speed $s_n$ and good rate function $\mathcal I$.
Then, if $\lbrace X_n, n\in \mathbb N\rbrace$ and $\lbrace Y_n, n\in \mathbb N\rbrace$ are exponentially equivalent at
speed $s_n$, the same LDP holds for $\lbrace Y_n, n\in \mathbb N\rbrace$.
\end{lemma}
A Moderate Deviation Principle (MDP) is a class of LDPs for families of random variables depending on the choice of certain scalings in a suitable class. Moreover, all these LDPs (whose speed function depends on the scaling) are ruled by the same quadratic rate function vanishing at zero (actually, usually one deals with families of centered random variables, or asymptotically centered).
In several cases the choice of the scaling parameters allows to fill the gap between the convergence in probability to a constant and the convergence in law to a centered Gaussian random variable.
\begin{example}\label{exMDP}
Let $\lbrace Z_n, n\in \mathbb N\rbrace$ be a sequence of i.i.d. real-valued random variables such that the moment-generating function
\begin{equation*}
\mathbb R \ni \lambda\mapsto   \log \mathbb E\left [\e^{ \lambda Z_1} \right ] \end{equation*}
is finite in some ball around the origin, $\mathbb E[Z_1]=0$ and $\Var(Z_1) = 1$. Let us define for each $n$ the random variable
\begin{equation*}
X_n :=  \sum_{i=1}^n Z_i/\sqrt{n}.
\end{equation*}
For any sequence of positive numbers $\lbrace a_n, n\in \mathbb N\rbrace$ such that
$$
a_n\to +\infty,\qquad a_n/\sqrt n \to 0,
$$
a MDP with speed $a_n^2$ and rate function $\mathcal I(x) := x^2/2, x\in \mathbb R$, holds for $\lbrace X_n, n\in \mathbb N\rbrace$, namely for every Borel set $B\subset \mathbb R$
\begin{equation*}
- \inf_{x\in \mathring{B}} \mathcal I(x)\le \liminf_{n\to +\infty} \frac{1}{a^2_n} \log \mathbb P(X_n/a_n\in B) \le \limsup_{n\to +\infty} \frac{1}{a^2_n} \log \mathbb P(X_n/a_n\in B) \le - \inf_{x\in \bar{B}} \mathcal I(x),
\end{equation*}
where $\mathring B$ (resp. $\bar B$) denotes the interior (resp. the closure) of $B$.
\end{example}
The classical example \cite[Theorem 3.7.1]{DZ98} recalled just above concerns the empirical means of i.i.d. random variables; indeed, if these random variables have finite moment generating function in a neighborhood of the origin, a class of LDPs holds filling the gap between the asymptotic regimes of the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT): indeed, using the same notation as in Example \ref{exMDP}, as $n\to +\infty$ we have
\begin{equation*}
X_n/\sqrt{n} \to 0\quad \mathbb P-a.s.,\qquad X_n \mathop{\to}^d Z\sim \mathcal N(0,1),
\end{equation*}
where $\mathop{\to}^d$ denotes convergence in distribution.

 Furthermore, for completeness, we recall that a LDP linked to a LLN is provided by the celebrated Cram\'er Theorem \cite[Theorem 2.2.3]{DZ98}.

\section{Main results and outline of the paper}

\subsection{Statement of main results}

Our main results concern MDPs which refine Theorems \ref{thS} and \ref{thB}, namely a class of LDPs for nodal lengths of random spherical harmonics for certain scalings $\{a_\ell,\ell\in\mathbb{N}\}$ that tend to infinity
slowly (see conditions (\ref{cond1}) and (\ref{cond2}) in Theorems \ref{th1} and \ref{th2} below), with speed
$a_\ell^2$, and common quadratic rate function $\mathcal I(x)=x^2/2$, $x\in \mathbb R$. Moreover, for
$a_\ell=1$ (and in such a case the condition $a_\ell \to +\infty$ in (\ref{cond1}) and (\ref{cond2}) fails) we have the convergence in law to the standard Normal distribution (Theorem \ref{thS} and Theorem \ref{thB}).
Recall the preliminaries
in Section \ref{sub:MDPs} and the discussion just after the statement of Lemma \ref{thDZ}.
\begin{theorem}\label{th1}
Let $\lbrace a_\ell, \ell \in \mathbb N\rbrace$ be any sequence of positive numbers such that, as $\ell\to +\infty$,
\begin{equation}\label{cond1}
a_\ell \to +\infty,\qquad \qquad a_\ell/\sqrt{\log \log \ell} \to 0.
\end{equation}
The sequence of random variables
$$\left \lbrace \widetilde{\mathcal L}_\ell/a_\ell, \ell \in \mathbb N \right \rbrace$$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x) = x^2/ 2$, $x\in \mathbb R$.
\end{theorem}
From now on,  $\lbrace r_\ell, \ell \in \mathbb N\rbrace$ is a sequence of radii such that (see Theorem \ref{thB})
$$r_\ell \ell \to +\infty.
$$
\begin{theorem}\label{th2}
Let $\lbrace a_\ell, \ell \in \mathbb N\rbrace$ be any sequence of positive numbers such that, as $\ell\to +\infty$,
\begin{equation}\label{cond2}
a_\ell \to +\infty,\qquad \qquad a_\ell/\sqrt{\log \log r_\ell \ell} \to 0.
\end{equation}
The sequence of random variables
$$\left \lbrace \widetilde{\mathcal L}_{\ell, B_{r_\ell}}/a_\ell, \ell \in \mathbb N \right \rbrace$$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x) = x^2/ 2$, $x\in \mathbb R$.
\end{theorem}
To the best of our knowledge, Theorem \ref{th1} and Theorem \ref{th2} are the first MD estimates for Lipschitz-Killing curvatures of excursion sets of Laplacian Gaussian eigenfunctions on manifolds and on shrinking domains on manifolds, respectively.

\begin{remark}[Lipschitz-Killing curvatures]\rm
The conditions $a_\ell/\sqrt{\log \log \ell}\to 0$ in (\ref{cond1}) and $a_\ell/ \sqrt{\log \log r_\ell \ell}\to 0$ in (\ref{cond2}) are plausibly not optimal; this is a drawback of the proof technique we decided to adopt. Our choice was based on the fact that we do not have any information on the moment-generating function of the nodal length and, more importantly, on the shortness of our argument as well as on its flexibility. Indeed, our strategy can be immediately adapted to prove MD estimates for the other Lipschitz-Killing (LK) curvatures of excursion sets at any level of random spherical harmonics such as the length of level curves, the excursion area and the Euler-Poincar\'e characteristic (see e.g. \cite{CM18, MW14, Tod19} and the references therein). Actually, our approach works well independently of the underlying manifold as long as the random model is Gaussian and long-memory, for instance in the case of arithmetic random waves, see \cite{BMW18, Cam19, MPRW16, NPR19, PR18} and the references therein.
\end{remark}

\subsection{On the proofs of the main results}

The nodal length $\mathcal L_\ell$ in (\ref{nodalDef}) is a finite-variance functional of the Gaussian field $T_\ell$ (Definition \ref{defRSH}) hence
it can be written as an orthogonal series, the so-called \emph{Wiener-\^Ito chaos expansion}, converging in $L^2(\mathbb P)$ of the form
\begin{equation}\label{exp1}
\mathcal L_\ell = \mathbb E[\mathcal L_\ell] + \sum_{q=1}^{+\infty} \mathcal L_\ell[q],
\end{equation}
where $\mathcal L_\ell[q]$ is the orthogonal projection of $\mathcal L_\ell$ onto the so-called Wiener chaos of order $q$. The expansion in (\ref{exp1}) relies on the fact that the family of suitably normalized Hermite polynomials $\lbrace H_q, q\in \mathbb N\rbrace$
is an orthonormal basis for the space of square integrable functions on the real line with respect to the Gaussian density. Recall that
$H_0 \equiv 1$ and
 \begin{equation}\label{hermite}
 H_{q}(t) := (-1)^q \phi^{-1}(t) \frac{d^q}{dt^q} \phi(t), \qquad t\in \mathbb R, q\in \mathbb N_{\ge 1},
 \end{equation}
 where $\phi$ denotes the standard Gaussian density. In particular $H_1(t)=1$, $H_2(t)=t^2-1$, $H_3(t)= t^3 -3t$ and $H_4(t) = 3t^4 - 6t^2 +3$.
It turns out that
 $\mathcal L_\ell - \mathbb E[\mathcal L_\ell]$ is asymptotically equivalent, in the $L^2(\mathbb P)$-sense, to its fourth chaotic component $\mathcal L_\ell[4]$
which is moreover fully correlated, as $\ell\to +\infty$, to the \emph{sample trispectrum} of $T_\ell$. More precisely
let us define, for $\ell\in \mathbb N$,
\begin{equation}\label{M}
\mathcal M_\ell := -\frac14 \sqrt{\frac{\ell(\ell+1)}{2}} \frac{1}{4!} \int_{\mathbb S^2} H_4(T_\ell(x))\,dx;
\end{equation}
Theorem 1.2 in \cite{MRW20} states that, as $\ell\to +\infty$,
\begin{equation}\label{L2equivalence}
\mathbb E \left [ \left | \widetilde{\mathcal L}_\ell -  \widetilde{\mathcal M}_\ell \right |^2 \right ] = O\left ((\log \ell)^{-1}\right ),
\end{equation}
where
$$\widetilde{\mathcal M}_\ell := \frac{\mathcal M_\ell}{\sqrt{\Var(\mathcal M_\ell)}}.$$

Now recall that the sequence of random variables $\lbrace \widetilde{\mathcal M}_\ell, \ell \in \mathbb N\rbrace$ lives in the fourth Wiener chaos. On that space, convergence in law to a standard Gaussian random variable can be proved  \cite{NP05, NP09} showing that its fourth cumulant goes to zero, hence
\begin{equation}\label{4cum}
\text{Cum}_4 ( \widetilde{\mathcal M}_\ell  )\to 0.
\end{equation}

Theorem 1 in \cite{ScTh16} ensures that under (\ref{4cum}) a MDP holds for the normalized sample trispectrum. See also \cite{DE13} for MDP via cumulants and \cite{Led90} for LDP for Wiener chaos.

In this paper, as a preliminary result (Lemma \ref{lem1}), firstly we will establish a MDP for $\lbrace \widetilde{\mathcal M}_\ell/a_\ell, \ell \in \mathbb N\rbrace$ with the rate function $\mathcal I$ in Theorem \ref{thS} and speed $a_\ell^2$, whenever as $\ell\to +\infty$
\begin{equation*}\label{a1}
a_\ell \to +\infty,\qquad \qquad a_\ell/(\log\ell)^{1/7}\to 0.
\end{equation*}
Then, in order to deduce a MDP for the whole series on the right hand side of (\ref{exp1}), i.e. to establish Theorem \ref{th1}, we will prove that $\lbrace \widetilde{\mathcal M}_\ell/a_\ell, \ell\in \mathbb N\rbrace$ and $\lbrace \widetilde{\mathcal L}_\ell/a_\ell, \ell \in \mathbb N	\rbrace$ are exponentially equivalent (Definition \ref{expdef}) at speed $a_\ell^2$ provided that $a_\ell$ goes to infinity sufficiently slowly (see (\ref{cond1})) according to (\ref{L2equivalence}). Finally Lemma \ref{thDZ} will allow to conclude.

The proof of Theorem \ref{th2} relies on the same ideas as those developed just before for the proof of Theorem \ref{th1}.

\subsection{Plan of the paper}

In \S \ref{secWiener} we recall both the notion of Wiener chaos and the chaotic expansion (\ref{exp1}) for the nodal length of random spherical harmonics. In particular, in \S \ref{sec_red} we recall the reduction principle for nodal lengths on the sphere and on shrinking domains leading in particular to (\ref{L2equivalence}). Finally in \S \ref{sec_proofs} we give the proofs of our main results.

\section{Nodal lengths and Wiener chaos}\label{secWiener}

\subsection{Wiener chaos}\label{sub_wiener}

It is well-known \cite[Proposition 1.4.2]{NP12} that the family $ \{H_q/\sqrt{q!}, q\in \mathbb N\}$ of suitably normalized Hermite polynomails (\ref{hermite}) is a complete orthonormal system in the space of square integrable real functions $L^2(\phi)$ with respect to the standard Gaussian measure on the real line.

Random spherical harmonics \eqref{defT} are linear combinations of i.i.d. standard Gaussian random variables $\{a_{\ell,m} : \ell = 1, 2, \dots, m=1, \dots, 2\ell+1\}$; we define accordingly the space $X$ to be the closure in $L^2(\mathbb{P})$ of $\text{lin}\{a_{\ell,m} : \ell = 1, 2, \dots, m=1, \dots, 2\ell+1\}$,
thus $X$ is a real centered Gaussian Hilbert subspace
of $L^2(\mathbb{P})$.
Now let $q\in \mathbb N$; the $q$-th Wiener chaos $C_q$ associated with $X$ is defined as the closure in $L^2(\mathbb{P})$ of all real finite linear combinations of random variables of the form
$$
H_{p_1}(x_1) H_{p_2}(x_2)\cdots H_{p_k}(x_k)
$$
for $k\in \mathbb N_{\ge 1}$, where $p_1,...,p_k \in \mathbb N$ satisfy $p_1+\cdots+p_k = q$, and $(x_1, x_2, \dots, x_k)$ is a standard Gaussian vector extracted
from $X$ (plainly, $C_0 = \mathbb{R}$ and $C_1 = X$). Note that (from (\ref{defT})) for every $\ell$ the random fields $T_{\ell}$ and $\nabla T_\ell$ viewed as collections of
Gaussian random variables
indexed by $x\in \mathbb S^2$ are all lying in $X$.

Taking into account the orthonormality and completeness of $\{H_q/\sqrt{q!}, q\in \mathbb N\}$ in $L^2(\phi)$, together with a monotone class argument (see e.g. \cite[Theorem 2.2.4]{NP12}), one can prove that $C_q \,\bot\, C_{q'}$ in  $L^2(\mathbb{P})$ whenever $q\neq q'$, and moreover
\begin{equation*}
L^2_X(\mathbb{P}) = \bigoplus_{q=0}^\infty C_q,
\end{equation*}
where $L^2_X(\mathbb P) := L^2(\Omega, \sigma(X), \mathbb P)$, that is, every finite-variance real-valued functional $F$ of $X$ admits a unique representation as a series, converging in $L^2_X(\mathbb P)$, of the form
\begin{equation}\label{e:chaos2}
F = \sum_{q=0}^\infty F[q],
\end{equation}
$F[q]:=\text{proj}(F \, | \, C_q)$ being the orthogonal projection of $F$ onto $C_q$ (in particular, $F[0]= \E [F]$).
 For a complete discussion on Wiener chaos see \cite[\S 2.2]{NP12} and the references therein.

\subsection{Nodal length: chaos expansion}

Let $B\subseteq \mathbb S^2$ be a ``nice" subset of the sphere. For our purpose it suffices to take $B$ as the whole sphere or a spherical cap. The nodal length $\mathcal L_{\ell, B} := \text{length}(T_\ell^{-1}(0) \cap B)$ in $B$ at frequency $\ell$ (plainly, $\mathcal L_{\ell, \mathbb S^2}  \equiv \mathcal L_\ell$)
can be formally written as
\begin{equation}\label{rapSphere}
\mathcal L_{\ell, B} = \int_{B} \delta_0(T_\ell(x)) \|\nabla T_\ell(x)\|\,dx,

\end{equation}
where $\delta_0$ stands for the Dirac mass in $0$, $\nabla T_\ell$ is the gradient field and $\| \cdot \|$ denotes the Euclidean norm in $\R^2$. Indeed, let us consider the $\eps$-approximating random variable
$$
\mathcal L_{\ell,B}^\eps := \frac{1}{2\eps} \int_{B} \chi_{[-\eps, \eps]}(T_\ell(x))\, \| \nabla T_\ell(x)\|\, dx,
$$
where $0 < \eps\ll 1$ and $\chi_{[-\eps, \eps]}$ denotes the indicator function of the interval $[-\eps, \eps]$. It is possible to prove that
\begin{equation*}
\lim_{\eps \to 0} \mathcal L_{\ell,B}^\eps = \text{length}(T_\ell^{-1}(0)\cap B)
\end{equation*}
both $\mathbb P-$a.s. and in $L^2(\P)$, see \cite{MRW20, NPR19}, thus justifying \paref{rapSphere}.
In particular, $\mathcal L_{\ell,B}\in L^2_X(\mathbb P)$.
The integral representation \paref{rapSphere} can be equivalently written as
\begin{equation}\label{rapSphere2}
\mathcal L_{\ell, B}
=\sqrt{\frac{\ell(\ell+1)}{2}}\int_{B} \delta_0(T_\ell(x)) \|\widetilde \nabla T_\ell(x)\|\,dx,
\end{equation}
where  $\widetilde \nabla$ is the normalized gradient, i.e. $\widetilde \nabla:= \nabla / \sqrt{\frac{2}{\ell(\ell +1)}}$ (thus pointwise the components of the normalized gradient have unit variance, see \S 3.2.1 in \cite{MRW20} for details).
Let us now recall the chaotic expansion \paref{e:chaos2} for $\mathcal L_{\ell, B}$
\begin{equation}\label{chaos_decomp}
\mathcal L_{\ell,B} = \sum_{q=0}^{+\infty} \mathcal L_{\ell,B}[2q],
\end{equation}
where $\mathcal L_{\ell,B}[2q]$ denotes the orthogonal projection of $\mathcal L_{\ell,B}$ onto  $C_{2q}$. (Note that projections on odd chaoses vanish since the integrand functions in \paref{rapSphere2} are both even.)
In \cite[$\S2$]{MRW20} the terms of the series on the right hand side of \paref{chaos_decomp}
are explicitly given (see also \cite{MPRW16}).
Let us introduce the two sequences of real numbers $\lbrace \beta_{2k}\rbrace_{k=0}^{+\infty}$ and $\lbrace \alpha_{2n,2m}\rbrace_{n,m=0}^{+\infty}$ corresponding to the (formal) chaotic coefficients of the Dirac mass at $0$ and the Euclidean norm respectively: for $k, n,m\in \mathbb N$
\begin{equation*}
\beta_{2k} := \frac{1}{\sqrt{2\pi}} H_{2k}(0),\qquad \qquad \alpha _{2n,2m}:=\sqrt{\frac{\pi }{2}}\frac{(2n)!(2m)!}{n!m!}\frac{1}{2^{n+m}
}p_{n+m}\left(\frac{1}{4}\right),
\end{equation*}
where $p_{N}$ is the swinging factorial coefficient
$
p_{N}(x):=\sum_{j=0}^{N}(-1)^{N+j}\left(
\begin{array}{c}
N \\
j
\end{array}
\right) \frac{(2j+1)!}{(j!)^{2}}x^{j}
$, $x\in \mathbb R$.
The $2q$-th chaotic projection of the nodal length restricted to $B$ is
\begin{equation}\label{chaos_decomp_sphere}
\begin{split}
 \mathcal L_{\ell,B} [2q] = &\sqrt{\frac{\ell (\ell
+1)}{2}} \sum_{u=0}^{q}\sum_{k=0}^{u}\frac{\alpha
_{2k,2u-2k}\beta _{2q-2u}}{(2k)!(2u-2k)!(2q-2u)!} \\
&\times \int_{B}H_{2q-2u}(T_{\ell }(x))H_{2k}(\widetilde \partial
_{1;x}T_{\ell }(x))H_{2u-2k}(\widetilde \partial
_{2;x}T_{\ell }(x))\,dx,
\end{split}
\end{equation}
where we use spherical coordinates (colatitude $\theta ,$ longitude $
\varphi $) and for $x=(\theta _{x},\varphi _{x})$ we are using the notation
\begin{equation*}
\widetilde \partial _{1;x}=\left. (\ell(\ell+1)/2)^{-1/2}\cdot \frac{\partial }{\partial \theta }\right\vert
_{\theta =\theta _{x}},\qquad \widetilde \partial _{2;x}= (\ell(\ell+1)/2)^{-1/2}\cdot\left. \frac{1}{\sin \theta }
\frac{\partial }{\partial \varphi }\right\vert _{\theta =\theta _{x},\varphi
=\varphi _{x}}.
\end{equation*}
Obviously $\mathcal L_{\ell,B}[0] = \frac{\text{area}(B)}{2\sqrt{2}} \sqrt{\ell(\ell+1)}  = \mathbb E[\mathcal L_{\ell,B}]$.

\subsection{Nodal lengths: reduction principles}\label{sec_red}

\subsubsection{On the sphere}\label{sub_sphere}

From (\ref{chaos_decomp_sphere}) for $q=1$, an application of Green's formula yields  (see \cite{MR19} and the references therein)
\begin{equation}
\mathcal L_\ell[2] = 0.
\end{equation}
Let us consider, as in (\ref{M}), the sample trispectrum
\begin{equation*}
\mathcal M_\ell := -\frac14 \sqrt{\frac{\ell(\ell+1)}{2}}\frac{1}{4!} \int_{\mathbb S^2} H_4(T_\ell(x))\,dx;
\end{equation*}
from the properties of Hermite polynomials recalled in \S \ref{sub_wiener} and by (\ref{hilb}) we have
\begin{equation}\label{eqVarM}
\mathbb E[\mathcal M_\ell] = 0, \qquad \qquad
\Var(\mathcal M_\ell) = \frac{1}{32} \log \ell + O(1), \text{ as } \ell\to +\infty
\end{equation} (see \cite[Lemma 3.2]{MW14}), cf. (\ref{eqVar}). Moreover, in \cite{MRW20} it has been shown that the fourth chaotic projection $\mathcal L_\ell[4]$ (see (\ref{chaos_decomp_sphere}) for $q=2$) is asymptotically (as $\ell\to +\infty$) equivalent, in the $L^2(\mathbb P)$-sense, to $\mathcal M_\ell$, and that the tail $\sum_{q\ge 3} \mathcal L_\ell[2q]$ of the chaotic series (\ref{chaos_decomp})  is negligible. To be more precise,
\begin{equation}\label{red1}
\mathbb E\left [ \left | \widetilde{\mathcal L}_\ell -\widetilde{\mathcal M}_\ell   \right |^2 \right ] = O\left ( (\log \ell)^{-1}\right ),\quad \text{ as } \ell\to+\infty,
\end{equation}
which is (\ref{L2equivalence}). An application of the Fourth Moment Theorem \cite{NP05, NP09} gives \cite{MW14}
\begin{equation*}
d_W \left ( \widetilde{\mathcal M}_\ell, Z\right ) \le \sqrt{\text{Cum}_4 (\widetilde{\mathcal M}_\ell  )} = O \left ( (\log \ell)^{-1} \right ),\qquad \text{ as } \ell\to +\infty,
\end{equation*}
that together with the estimate (\ref{red1}) proves (\ref{Wass1}).

\subsubsection{On shrinking spherical domains} \label{shrinking}

Let us define the local sample trispectrum as
\begin{equation}\label{Mloc}
\mathcal M_{\ell,B_{r_\ell}} := -\frac14 \sqrt{\frac{\ell(\ell+1)}{2}}\frac{1}{4!} \int_{B_{r_\ell}} H_4(T_\ell(x))\,dx.
\end{equation}
It has zero mean and variance (recall that $r_\ell \ell\to +\infty$) given by
\begin{equation}\label{eqVarL}
\Var(\mathcal M_{\ell,B_{r_\ell}}) = \frac{1}{256}r_\ell^2 \log r_\ell \ell + O(r_\ell^2)
\end{equation}
 (see \cite[Proposition 3.4]{Tod18}).
In \cite[Proposition 3.3]{Tod18} the asymptotic full correlation between the local nodal length $\mathcal{L}_{\ell,B_{r_\ell}}$ and the local sample trispectrum has been established and, in view of the orthogonality of the projections, this entails that $\mathcal{M}_{\ell,B_{r_\ell}}$ is the leading term of the chaos expansion of $\mathcal{L}_{\ell,B_{r_\ell}}$ in (\ref{chaos_decomp_sphere}). Indeed, all the other projections are proved to be $O(r_\ell^2)$ in the $L^2-$sense (see the supplement article to \cite{Tod18}, Appendix C) and hence we can conclude that
\begin{equation}\label{L2palla}
\mathbb E\left [ \left | \widetilde{\mathcal L}_{\ell,B_{r_\ell}} -\widetilde{\mathcal M}_{\ell,B_{r_\ell}}   \right |^2 \right ] = O\left ( (\log r_\ell \ell)^{-1}\right ).
\end{equation}
Hence the Fourth Moment Theorem \cite{NP05, NP09} gives \cite[Lemma 5.4]{Tod18}
\begin{equation*}
d_W \left ( \widetilde{\mathcal M}_{\ell, B_{r_\ell}}, Z\right )  \le \sqrt{\text{Cum}_4 (\widetilde{\mathcal M}_{\ell, B_{r_\ell}}  )} = O \left ( (\log r_\ell \ell)^{-1/2} \right )
\end{equation*}
that together with the estimate (\ref{L2palla}) proves (\ref{Wass2}).

\section{Proofs of the main results}\label{sec_proofs}

\subsection{Proof of Theorem \ref{th1}}

Bearing in mind \S \ref{sub_sphere} we start with an auxiliary lemma of independent interest which provides the MDP for the sample trispectrum on the sphere.

\begin{lemma}\label{lem1}
Let $\lbrace a_\ell, \ell\in \mathbb N\rbrace$ be any sequence of positive numbers such that, as $\ell\to +\infty$,
\begin{equation}\label{condM}
a_\ell \to +\infty,\qquad \qquad a_\ell/(\log\ell)^{1/7}\to 0.
\end{equation}
Then the sequence of random variables $\lbrace \widetilde{\mathcal M}_\ell/ a_\ell, \ell\in \mathbb N\rbrace$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x) =x^2/2$, $x\in \mathbb R$.
\end{lemma}
\begin{proof}
The random variable $\mathcal M_\ell$ belongs to the fourth Wiener chaos $C_4$ for each $\ell\in \mathbb N$.
Recalling (\ref{eqVarM}),
from \cite[Lemma 3.3]{MW14}, we have that, as $\ell\to +\infty$, $\text{Cum}_4(\mathcal M_\ell) \approx 1$.
Hence we have, as $\ell\to +\infty$,
\begin{equation}\label{cum}
\text{Cum}_4(\widetilde{\mathcal M}_\ell) = \frac{\text{Cum}_4(\mathcal M_\ell)}{\Var(\mathcal M_\ell)^2} \approx \frac{1}{\log^2\ell}.
\end{equation}
Now let $a_\ell$ be a positive sequence such that, as $\ell \to +\infty$,
\begin{equation}\label{cumM}
a_\ell \to +\infty,\qquad \qquad \qquad a_\ell/ (\Delta_\ell)^{1/3} \to 0,
\end{equation}
where
\begin{equation}\label{Delta}
\Delta_\ell := \left ( \sqrt{\text{Cum}_4  (\widetilde{\mathcal M}_\ell )}\right )^{-3/7} \approx (\log \ell)^{3/7}.
\end{equation}
Note that (\ref{cumM}) is equivalent to (\ref{condM}).
Corollary 2 in \cite{ScTh16} ensures that the sequence $\widetilde{\mathcal M}_\ell/a_\ell$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x) = x^2/ 2$, $x\in \mathbb R$.

\end{proof}
\begin{remark}\label{remCP}\rm
Our argument leads to further results, namely expansions \`a la Cram\'er-Petrov \cite{RSS78} for the sample trispectrum. We state here the result in a simplified form taken from Theorem 5 (i) in \cite{ScTh16}: there exist universal constants $c_0, c_1, c_2 >0$ such that for $\Delta_\ell \ge c_0$, $0\le x \le c_1 \Delta_\ell^{1/3}$ ($\Delta_\ell$ is defined as in (\ref{Delta})) it holds that
\begin{equation*}
\left | \log \frac{\mathbb P(\widetilde{\mathcal M}_\ell \ge x)}{1 - \Phi(x)} \right | \le c_2 \frac{1+x^3}{\Delta_\ell^{1/3}},\qquad \left |  \log \frac{\mathbb P(\widetilde{\mathcal M}_\ell \le -x)}{ \Phi(-x)} \right |  \le c_2 \frac{1+x^3}{\Delta_\ell^{1/3}},
\end{equation*}
where $\Phi$ denotes the standard Gaussian cumulative distribution function. An anonymous referee raised the interesting question whether expansions \`a la Cram\'er-Petrov hold for nodal lengths on the sphere and in shrinking spherical domains. We leave this as a topic for future research.
\end{remark}

\begin{proof}[Proof of Theorem \ref{th1}]
We want to combine Theorem \ref{thDZ} and Lemma \ref{lem1} for every speed $a_\ell$ satisfying (\ref{cond1}); note that $a_\ell/\sqrt{\log \log \ell} \to 0$ implies $a_\ell/(\log \ell)^{1/7}\to 0$ (see (\ref{condM})). In order to do so, we have to check that for every $\delta >0$
\begin{equation}\label{limsup}
\limsup_{\ell\to +\infty}\frac{1}{a_\ell^{2}} \log \mathbb P( a_\ell^{-1} |  \widetilde{\mathcal {L}}_\ell -  \widetilde{\mathcal M}_\ell | > \delta) = -\infty,
\end{equation}
i.e., that the sequences of random variables $\lbrace  \widetilde{\mathcal L}_\ell /a_\ell, \ell\in \mathbb N\rbrace$ and $\lbrace  \widetilde{\mathcal {M}}_\ell/a_\ell, \ell\in \mathbb N\rbrace$ are exponentially equivalent (Definition \ref{expdef}) at speed $a_\ell^2$.
Thanks to Markov inequality we have
\begin{equation*}
\begin{split}
\mathbb P( a_\ell^{-1} |  \widetilde{\mathcal L}_\ell  -  \widetilde{\mathcal M}_\ell | > \delta) \le \frac{a_\ell^{-2} \mathbb E[| \widetilde{\mathcal L}_\ell  -  \widetilde{\mathcal M}_\ell |^2]}{\delta^2}
\end{split}
\end{equation*}
so that
\begin{equation}\label{app1}
\begin{split}
\limsup_{\ell\to +\infty}a_\ell^{-2} \log \mathbb P( a_\ell^{-1} |   \widetilde{\mathcal L}_\ell -  \widetilde{\mathcal M}_\ell | > \delta) &\le \limsup_{\ell\to +\infty}a_\ell^{-2} \log \frac{a_\ell^{-2} \mathbb E[| \widetilde{\mathcal L}_\ell  -  \widetilde{\mathcal M}_\ell |^2]}{\delta^2}\cr
&= \limsup_{\ell\to +\infty}a_\ell^{-2} \log  \mathbb E[| \widetilde{\mathcal L}_\ell  -  \widetilde{\mathcal M}_\ell |^2].
\end{split}
\end{equation}
Plugging (\ref{L2equivalence}) into (\ref{app1}) we get (\ref{limsup}) whenever $a_\ell = o(\sqrt{\log \log \ell})$ as in (\ref{cond1}).

\end{proof}

\subsection{Proof of Theorem \ref{th2}}
Here we refer to \S \ref{shrinking} and we follow the same lines of the proof of Theorem \ref{th1}. We first prove the following lemma which is of independent interest.

\begin{lemma}\label{lem1bis}
	Let $\lbrace a_\ell, \ell\in \mathbb N\rbrace$ be any sequence of positive numbers such that, as $\ell\to +\infty$,
	\begin{equation}\label{condAP}
	a_\ell \to +\infty,\qquad \qquad a_\ell/(\log r_\ell \ell)^{1/14} \to 0.
	\end{equation}
	Then the sequence of random variables $\lbrace \widetilde{\mathcal M}_{\ell, B_{r_\ell}}/ a_\ell, \ell\in \mathbb N\rbrace$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x)=x^2/2$, $x\in \mathbb R$.
\end{lemma}
\begin{proof}
	The random variable $\mathcal{M}_{\ell, {B_{r_\ell}}}$ in (\ref{Mloc}) belongs to the fourth Wiener chaos for each $\ell\in \mathbb N$. Recalling (\ref{eqVarL}) and that from \cite[Lemma 5.4]{Tod18}, as $\ell\to +\infty$, $\text{Cum}_4(\mathcal{M}_{\ell, B_{r_\ell}}) =O( r_\ell^4 \log r_\ell \ell)$,
 we have
	\begin{equation}\label{cumbis}
	\text{Cum}_4(\widetilde{\mathcal M}_{ \ell, B_{r_\ell}}) = \frac{\text{Cum}_4(\mathcal{M}_{\ell, B_{r_\ell}})}{\Var(\mathcal{M}_{\ell, B_{r_\ell}})^2} =O\left ( \frac{1}{\log r_\ell \ell} \right ).
	\end{equation}
	Now let $a_\ell$ be a positive sequence such that, as $\ell \to +\infty$,
	\begin{equation}\label{cumAP}
	a_\ell \to +\infty,\qquad \qquad \qquad a_\ell/ (\Delta_{\ell, r_\ell})^{1/3} \to 0,
	\end{equation}
	where
	$$
	\Delta^{-1}_{\ell, r_\ell} := \left ( \sqrt{\text{Cum}_4 (\widetilde{\mathcal M}_{ \ell, B_{r_\ell}}  )}\right )^{3/7} = O\left (  (\log r_\ell \ell)^{-3/14} \right),\ \text{ as } \ell\to +\infty.
	$$
	Note that (\ref{condAP}) implies (\ref{cumAP}). It follows that,
as in the proof of Lemma \ref{lem1}, Corollary 2 in \cite{ScTh16} ensures that the sequence $\lbrace \widetilde{\mathcal M}_{\ell,B_{r_\ell}}/a_\ell, \ell \in \mathbb N\rbrace$ satisfies a MDP with speed $a_\ell^2$ and rate function $\mathcal I(x) = x^2/ 2$.

\end{proof}

Analogous results as those in Remark \ref{remCP} hold for the sample trispectrum restricted to a shrinking ball.

\begin{proof}[Proof of Theorem \ref{th2}] Along the same lines of the proof of Theorem \ref{th1} we want to combine Theorem \ref{thDZ} and Lemma \ref{lem1bis} for every speed $a_\ell$ satisfying (\ref{cond2}); note that $a_\ell/\sqrt{\log \log r_\ell \ell} \to 0$ implies $a_\ell/(\log r_\ell \ell)^{1/14}\to 0$ (see (\ref{condAP})). To this aim, we have to check that for every $\delta >0$
	\begin{equation}\label{limsupbis}
	\limsup_{\ell\to +\infty}\frac{1}{a_\ell^{2}} \log \mathbb P( a_\ell^{-1} |  \widetilde{\mathcal{L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}} | > \delta) = -\infty.
	\end{equation}
Applying Markov inequality we have
	\begin{equation*}
	\begin{split}
	\mathbb P( a_\ell^{-1} |  \widetilde{\mathcal{L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}} | > \delta) \le \frac{a_\ell^{-2} \mathbb E[| \widetilde{\mathcal {L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}} |^2]}{\delta^2}
	\end{split}
	\end{equation*}
and hence
	\begin{equation}\label{app4}
	\begin{split}
	\limsup_{\ell\to +\infty}a_\ell^{-2} \log \mathbb P( a_\ell^{-1} |   \widetilde{\mathcal {L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}} | > \delta) &\le \limsup_{\ell\to +\infty}a_\ell^{-2} \log \frac{a_\ell^{-2} \mathbb E[| \widetilde{\mathcal {L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}} |^2]}{\delta^2}\cr
	&= \limsup_{\ell\to +\infty}a_\ell^{-2} \log  \mathbb E[| \widetilde{\mathcal {L}}_{\ell,B_{r_\ell}} -  \widetilde{\mathcal M}_{\ell,B_{r_\ell}}|^2].
	\end{split}
	\end{equation}
	Using (\ref{L2palla}) in (\ref{app4}) and taking $a_\ell = o(\sqrt{\log \log r_\ell	\ell})$, (\ref{limsupbis}) holds.

\end{proof}

\subsection*{Acknowledgements}

The authors would like to thank an anonymous referee for his/her valuable comments which improved the quality of the present work.
C.M. has been supported by MIUR Excellence Department Project awarded to the Department of Mathematics,
University of Rome Tor Vergata (CUP E83C18000100006) and by GNAMPA-INdAM (project:
\emph{Stime asintotiche: principi di invarianza e grandi deviazioni}). M.R. has been supported by
GNAMPA-INdAM (project: \emph{Propriet\`a analitiche e geometriche di campi aleatori}) and the
ANR-17-CE40-0008 project \emph{Unirandom}. A.P.T. has been funded by the German Research
Foundation (DFG) via RTG 2131 and by GNAMPA-INdAM (project:
\emph{Stime asintotiche: principi di invarianza e grandi deviazioni}).


text:
\begin{document}
\maketitle

\begin{abstract}
Communication systems for multicasting information and energy simultaneously to more than one user are investigated.
In the system under study, a transmitter sends the same message and signal to multiple receivers over distinct and independent channels.  The fundamental communication limit under a received energy constraint, called the multicast capacity-energy function, is studied and a single-letter expression is derived.  This is based on coding theorems for compound channels.
The problem of receiver segmentation, where receivers are divided into related groups, is also considered.
\end{abstract}

\section{Introduction}
Over the last decade, energy harvesting communication systems have not only been an attractive research topic, but have also seen considerable progress towards practical design and implementation.
Wireless power delivery, rather than supplying power via wire or battery, may
save significant cost, especially where battery replacement is difficult \cite{TongLWZ2010,LeeLL2012}.

The idea of \emph{simultaneous} information/energy transmission (SIET) to an energy harvesting receiver provides the possibility of trading between information transfer and energy delivery \cite{ZhangMH2015,ClerckxZSNKP2018}.
There is a fundamental limit for point-to-point communication, when the transmitter must also transfer some minimal amount of energy to a receiver, as characterized by a \emph{capacity-energy} function \cite{Varshney2008} e.g.\ for discrete memoryless channels (DMCs) and Gaussian channels.  This capacity-energy function can also be computed for SIET over noisy coupled-inductor circuits, modeled as a frequency-selective channel with additive white Gaussian noise \cite{GroverS2010}. Suboptimal strategies such as time-switching, where the transmitter switches between information delivery and energy delivery phases, can also be characterized \cite{Varshney2012,LiuZC2013}. In \cite{TandonMV2016}, the constrained capacity over DMCs was derived when each \emph{subblock} in a codeword must carry sufficient energy to avoid receiver energy outage. These subblock energy-constrained codes were generalized to \emph{skip-sliding window codes} in \cite{WuTVM2017}.

There have been numerous multiterminal extensions to the basic energy/information transmission problem \cite{FouladgarS2012,ZhangH2013}.
In \cite{FouladgarS2012}, SIET over multiple access channels was analyzed, whereas
\cite{ZhangH2013} studied broadcasting information/energy messages from a base station to energy-harvesting nodes and information-retrieving nodes.
Previous work has, however, not studied the practically important \emph{multicast} setting of Fig.~\ref{fig:multicast}, where the same message is transmitted over distinct and independent channels.

Although the idea of multicasting PoWiFi has become prominent in the communication systems literature \cite{TallaKRNSG2017} (leading to much public attention
and even a start-up company), these ideas have not previously been tied to fundamental principles concerning physical layer capacity.  Note that beacon signals form a significant bulk of wireless protocols, and so the multicast problem (as compared to the broadcast problem) is often present.

\begin{figure}[t]
  \centering
    \includegraphics[width=3.4in]{multicast.png}\vspace{-0.1in}
  \caption{The multicast setting: communicating the same signal over distinct noisy channels to several energy-harvesting receivers.}
  \vspace{-0.15in}
  \label{fig:multicast}
\end{figure}

Tree-structured multicast is simply a compound channel, since requiring reliable
communication over several channels is equivalent to requiring reliable communication over an unknown channel.
Recall the capacity of a compound DMC does not increase if the decoder (but not the encoder) knows which channel is chosen from a given family of channels \cite{LapidothN1998}.  In defining compound capacity
here, in addition to requiring a small error probability over each channel, we also require a certain minimum energy delivery.

In SIET communication systems, one may consider partitioning receivers into groups, based on their distance from the base-station\cite{ZengCZ2017} or energy requirements.
Therefore, beyond determining the compound capacity-power function, we also consider \emph{receiver segmentation},
where we group channels into different categories, so the same signaling scheme is used for members of the same category, as in Fig.~\ref{fig:multicast_group}.
This is equivalent to optimally quantizing the parameter space of the compound channel under given
information/energy requirements.

\begin{figure}[t]
  \centering
    \includegraphics[width=3.5in]{multicast_group.png}\vspace{-0.1in}
  \caption{Communicating the same message and a few signals over distinct noisy channels to grouped energy harvesting receivers, the segmentation problem.}
  \vspace{-0.15in}
  \label{fig:multicast_group}
\end{figure}

In summary, this work defines energy/information multicast communication and derives a single-letter expression for its capacity under a received energy constraint. In addition, a brief discussion of the segmentation problem is presented.

\section{Point-to-Point Capacity-Energy Function}
Consider a DMC with transition probability $p_{Y|X}(x|y)$, in which
$x$ and $y$ are drawn from the input alphabet $\cX$ and the output alphabet $\cY$ with distribution $p_{X}(x)$ and $p_{Y}(y)$, respectively.
Let $b(y)$ be an energy function that maps a channel output $y\in\cY$ into a nonnegative real value.
When channel inputs described by random variables $X_1^n=(X_1, X_2, \ldots, X_n)$ are transmitted through the channel,
the corresponding channel outputs described by random variables $Y_1^n=(Y_1, Y_2, \ldots, Y_n)$
contain the average energy
\begin{equation}
E[b(Y_1^n)]=\sum_{y_1^n = (y_1,y_2,\ldots,y_n)\in \cY^n}b(y_1^n)p_{Y_1^n}(y_1^n),
\end{equation}
where $b(y_1^n)\triangleq\sum_{i=1}^n b(y_i)$.

Consider the point-to-point case:
an $n$-letter transmission which guarantees at least expected energy $nB$ being received
can be evaluated by the $n$th capacity-energy function as
\begin{equation}
C_n(B)=\max_{X_1^n:E[b(Y_1^n)]\ge nB}I(X_1^n;Y_1^n).
\end{equation}
As mentioned in \cite{Varshney2008},
the capacity of the channel under constraint $(b, B)$, denoted $C(B)$,
can be single-letterized as
\begin{equation}
C(B)=C_1(B).
\end{equation}

\section{Multicasting Capacity-Energy Function}
Let us consider the multicast SIET setup in Fig.~\ref{fig:multicast}.
The message alphabet $\cW$ is encoded by the function $f$, which maps each message word $w\in \cW$
into a codeword $x_1^n=f(w)\in\cX^n$.
The source broadcasts the generated codeword to $L$ distinct receivers over independent DMCs and the $\ell$th client receives $y(\ell)_1^n\in\cY(\ell)^n$
based on the transition probability $p_{Y(\ell )_1^n|X_1^n}(y(\ell)_1^n|x_1^n)$.
Since the channel is memoryless, $p_{Y(\ell )_1^n|X_1^n}(y(\ell)_1^n|x_1^n)=\prod_{i=1}^n p_{Y(\ell )|X}(y(\ell)_i|x_i)$.
The $\ell$th client decodes $y(\ell)_1^n$ into $\hat{W}_{\ell}\in\cW$ by its own decoder $g_\ell(\cdot )$ and
an error occurs when $\hat{W}_\ell\neq W$.
Let $\Theta=\{1,2,\ldots,L\}$ be the set of $L$ receivers.
The notion of error probability follows.
\begin{defin}
The error probability for the multicast channel, denoted by $\eta$, is defined
as the worst average error probability over all channels:
\begin{equation}
\eta=\max_{\ell\in\Theta}\Pr[W\neq\hat{W}_\ell].
\end{equation}
\end{defin}

The $\ell$th client receives energy $E[b_{\ell}(Y(\ell)_1^n)]$ on average,
where the energy harvesting function $b_{\ell}(\cdot)$ may be different for different receivers.
The energy constraint vector $\vec{B}=(B_1,B_2,\ldots,B_L)$ is an $L$-tuple vector,
which defines the least energy requirement for each channel, i.e.,
\begin{equation}\label{eqn:energy-constraint}
E[b_{\ell}(Y(\ell)_1^n)]\ge nB_\ell, \, \forall \ell\in\Theta.
\end{equation}
An input that satisfies \eqref{eqn:energy-constraint} is said to be $\vec{B}$-admissible.
Now we define the operational compound capacity in terms of an energy constraint vector $\vec{B}$.
\begin{defin}
An encoder $f$ maps a message $w\in\cW$ into a block of length $n$ with a rate $R=\frac{1}{n}\log |\cW|$.
Given $0\leq \epsilon < 1$,
a non-negative rate $R$ is said to be an $\epsilon$-achievable rate for multicast channels $\{p_{Y(\ell )|X}\}_{\ell\in\Theta}$
with an energy constraint vector $\vec{B}$ if
there exists a block code with error probability $\eta<\epsilon$ such that \eqref{eqn:energy-constraint} holds
and its rate exceeds $R-\delta$ for all $\delta>0$ when $n$ is sufficiently large.
$R$ is an achievable rate if it is $\epsilon$-achievable for all $\epsilon>0$.
The supremum of the achievable rate is called the operational capacity of the multicast channel, denoted $C_{\mathrm{O}}(\vec{B})$.
\end{defin}

To evaluate the capacity, let us consider informational quantities.
Let all feasible channel input distributions of dimension $n$ with the $\ell$th energy constraint be $\cF_n(\ell)$, i.e.,
\begin{equation}
\cF_n(\ell)=\left\{F_{X^n_1}: E[b_{\ell}(Y(\ell)_1^n)]\ge nB_\ell \right\},
\end{equation}
then the set all $\vec{B}$-admissible inputs is $\cF_n=\bigcap_{\ell\in\Theta}\cF_n(\ell)$.
The multicast capacity-energy function is then defined as follows.
\begin{defin}
Given a positive integer $n$ and an energy constraint vector $\vec{B}$, the $n$th multicast capacity-energy function $C_n(\vec{B})$ is defined as
\begin{equation}\label{eqn:nth-cefunction}
C_n(\vec{B})=\max_{X_1^n\in\cF_n}\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n),
\end{equation}
where $I(\cdot;\cdot)$ is the mutual information.
The domain ${\cal B}$ of $C_n(\vec{B})$ is all feasible energy constraints:
\begin{multline}\label{def:domain}
{\cal B}\triangleq \Bigg\{(B_1,\ldots ,B_L):  B_\ell=\vec{q} \cdot [P(\ell)] \cdot \vec{b}_\ell \quad \forall\ell \in \Theta\\
\mbox{ for all nonnegative }\vec{q}=\left[q_1\ \cdots\ q_{|\cX|}\right] \mbox{ s.t. } \sum_{i=1}^{|\cX|}q_i=1\Bigg\},
\end{multline}
where $\vec{b}_\ell$ is a column vector of the $[b_{\ell}(y)]_{y\in\cY}$ and $[P(\ell)]$ is the transition matrix $[p_{Y(\ell)|X}(y|x)]_{(x,y)\in\cX\times\cY}$.
The supremum rate of \eqref{eqn:nth-cefunction} defines the multicast capacity-energy function as
\begin{equation}\label{eqn:cefunction}
C(\vec{B})=\sup_n \frac{1}{n}C_n(\vec{B}).
\end{equation}
\end{defin}
Now we have a coding theorem.

\begin{thm}\label{thm:capacityfunction}
$C_{\mathrm{O}}(\vec{B})=C(\vec{B})$.
\end{thm}
\begin{IEEEproof}
Multicast energy/information communication can be treated as a compound channel with a constrained set of possible input distributions $\cF_n$.
Hence, the achievability and converse can be derived from random coding arguments and Fano's inequality, respectively.
See \cite[Thm.\ 7.1 \& Rem.\ 7.3]{ElGamalK2011}.
\end{IEEEproof}

Just as with traditional compound capacity, the compound capacity-energy function is not generally equal to the minimum of the individual
capacity-energy functions of the channels constituting the set since different channels may have different optimal input distributions.
That minimum is, however, an upper bound on the compound capacity-energy function.
Exchanging minimum and maximum operations is possible when the optimal input
distributions in the set are identical.

Although the multicast capacity-energy function provides an explicit form for capacity,
finding the supremum of $n$-dimensional mutual information is difficult.
Here we aim to provide a single-letter expression for the capacity.

For notational convenience, we consider the setting where the energy harvesting functions $b_{\ell}(\cdot)$
are all identically $b(\cdot)$ and all receivers have the same energy constraint $B$:
$B=B_1=B_2=\cdots=B_L\leq B_{\max}$,
where $B_{\max}$ is given as
\begin{equation}
B_{\max}=\max_{\substack{\vec{q}=\left[q_1\ \cdots\ q_{|\cX|}\right] \\\mbox{ s.t. } \sum_{i=1}^{|\cX|}q_i=1\, \&\, q_i\ge 0\,\forall i}}\min_{\ell}\vec{q}\cdot [P(\ell)]\cdot\vec{b}.
\end{equation}
Then $C_n(B)$ is a function of a scalar rather than a vector.  Extension to the general case follows
directly.

It is trivial that $C_n(B)$ is non-increasing along $B$.
The following theorem also shows that $C_n(B)$ is concave.

\begin{thm}\label{thm:concavity}
$C_n(B)$ is a concave function of $B$ for $0\le B\le B_{\max}$.
\end{thm}
\begin{IEEEproof}
Let $(X_i)_1^n$ and $\{Y_i(\ell)_1^n\}_{\ell\in\Theta}$ be the channel input of length $n$ and the corresponding channel outputs
which achieves $C_n(B_i)$ for $i=\{1,2\}$.
By definition, $E[b(Y_i(\ell)_1^n)]\ge nB_i$ for every $i$ and $\ell$.
Let $X_1^n$ be the channel input with the distribution $\alpha p_{(X_1)_1^n}+\beta p_{(X_2)_1^n}$ and $\{Y(\ell)_1^n\}_{\ell\in\Theta}$
be the corresponding channel outputs for some $\alpha\in[0,1]$ and $\beta=1-\alpha$.
The energy received from $Y(\ell)_1^n$ for the $\ell$th channel can be lower-bounded as
\begin{IEEEeqnarray}{rCl}
E[b(Y(\ell)_1^n)]&=&\alpha E[b(Y_1(\ell)_1^n)]+\beta E[b(Y_1(\ell)_1^n)]\\
&\ge& \alpha nB_1+\beta nB_2=n(\alpha B_1+\beta B_2).
\end{IEEEeqnarray}
Therefore, the channel input $X_1^n$ is $(\alpha B_1+\beta B_2)$-admissible.
By definition, we have the following inequalities for all $\ell\in\Theta$.
\begin{equation}
C_n(\alpha B_1+\beta B_2)\ge\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n).\label{eqn:concave-1}
\end{equation}
Since the mutual information is a concave function of the input distribution,
\begin{multline}\label{eqn:concave-2}
I(X_1^n;Y(\ell)_1^n)\ge\\
\alpha I((X_1)_1^n;Y_1(\ell)_1^n)+\beta I((X_2)_1^n;Y_2(\ell)_1^n)
\end{multline}
for all $\ell\in\Theta$.
Combining \eqref{eqn:concave-1} and \eqref{eqn:concave-2} validates the concavity:
\begin{align}
&C_n(\alpha B_1+\beta  B_2)\\
&\ge \min_{\ell\in\Theta}\left\{\alpha I((X_1)_1^n;Y_1(\ell)_1^n)+\beta I((X_2)_1^n;Y_2(\ell)_1^n)\right\}\\
&\ge \alpha \min_{\ell\in\Theta}\left\{I((X_1)_1^n;Y_1(\ell)_1^n)\right\}+\beta \min_{\ell\in\Theta}\left\{ I((X_2)_1^n;Y_2(\ell)_1^n)\right\}\\
&= \alpha C_n(B_1)+\beta C_n(B_2).
\end{align}
\end{IEEEproof}

The following theorem shows that the $n$th multicast capacity-energy function can be evaluated as a single-letter expression.

\begin{thm}\label{thm:letterization}
For any DMC, $C_n(B)=nC_1(B)$ for all $n\in\mathbb{Z}^+$ and $0\le B\le B_{\max}$.
\end{thm}
\begin{IEEEproof}
To prove the equality, we start with the case of ``$\le$''.
Let $X_1^n$ be the channel input with the corresponding outputs $\{Y(\ell)_1^n\}_{\ell\in\Theta}$ which achieves $C_n(B)$,
then we have $E[b(Y(\ell)_1^n)]\ge nB$ for all $\ell\in\Theta$.
We also let $X^*$ be the channel input with the corresponding outputs $\{Y^*(\ell)\}_{\ell\in\Theta}$ such that
$p_{X^*}=\sum_{i=1}^n \frac{1}{n}p_{X_i}$.
Due to the memoryless, We have
\begin{IEEEeqnarray}{rCl}
E[b(Y^*(\ell))]&=&\frac{1}{n}\sum_{i=1}^nE[b(Y(\ell)_i)]\\
&=&\frac{1}{n}E[b(Y(\ell)_1^n)]\ge B
\end{IEEEeqnarray}
and $I(X_1^n;Y(\ell)_1^n)\le\sum_{i=1}^nI(X_i;Y(\ell)_i)$ for all $\ell\in\Theta$.
Hence
\begin{IEEEeqnarray}{rCl}
C_n(B)&=&\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n)\\
&\le&\min_\ell \sum_{i=1}^nI(X_i;Y(\ell)_i)\\
&=&n\min_\ell \frac{\sum_{i=1}^nI(X_i;Y(\ell)_i)}{n}\\
&\le&n\min_\ell I(X^*;Y^*(\ell))\label{eqn:singleletter-1}\\
&\le&nC_1(B),\label{eqn:singleletter-2}
\end{IEEEeqnarray}
where \eqref{eqn:singleletter-1} follows from \eqref{eqn:concave-2} and \eqref{eqn:singleletter-2} follows from the definition.

For the case of ``$\ge$'',
we let $X$ be the channel input with the distribution $p_X$ and the corresponding outputs $\{Y(\ell)\}_{\ell\in\Theta}$ achieving $C_1(B)$.
A sequence of channel inputs $X_1^n=(X_1,X_2,\cdots,X_n)$ is created, in which each $X_i$ is i.i.d. to $X$.
Let $\{Y(\ell)_i^n\}_{\ell\in\Theta}$ be the corresponding channel outputs of $X_i^n$.
By definition, $E[b(Y(\ell)_1^n)]=\sum_{i=1}^nE[b(Y(\ell))]\ge nB$.
Therefore
\begin{IEEEeqnarray}{rCl}
C_n(B)&\ge&\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n)\\
&=&\min_{\ell\in\Theta}\sum_{i=1}^nI(X;Y(\ell))\\
&=&nC_1(B).
\end{IEEEeqnarray}
\end{IEEEproof}

Combining Theorems \ref{thm:capacityfunction} and \ref{thm:letterization} shows
the capacity of the multicast SIET as in Fig.~\ref{fig:multicast} is single-letterized as
\begin{equation}\label{eqn:capacity}
C_{\mathrm{O}}(B)=C_1(B).
\end{equation}

Although single-letterization was shown for identical energy harvesting functions and constraints,
the derivations are still valid for unequal constraints\footnote{
The single-letterization for unequal constraints $\vec{B}$ and different energy functions $b_\ell$
is given in Appendix \ref{append:letterization_general}}.

\subsection{Numerical Example}
The multicast capacity-energy function is evaluated for an example.
Considering a case of $L=2$, in which both channels are binary, denoted by $\Theta=\{\mathrm{BSC},\mathrm{Z}\}$.
The first channel is a binary symmetric channel (BSC) with crossover probability $\epsilon$;
the second channel is a Z-channel with a $1$ to $0$ probability $\epsilon_0$.
Let the energy function $b(\cdot)$ be identically defined as the Hamming weight of the output
and have common energy delivery requirement $B$.

Let $X^*(B)$ be the common capacity-achieving input distribution, whose computation is detailed elsewhere \cite{WuTVM2018_arXiv}.  For given values of $\epsilon$ and $\epsilon_0$, Fig.~\ref{fig:ex} individually plots
the mutual informations of both channels with input $X^*(B)$.
Since the multicast capacity is dominated by the worst channel, $C_{\mathrm{O}}(B)$ is simply the minimum of the curves.

\begin{comment}
Given an input with a distribution $p_X$,
the average energies received in two channels are $E[b(Y(\mathrm{BSC}))]=p_{Y(\mathrm{BSC})}(1)=p_X(0)\epsilon+p_X(1)(1-\epsilon)$ and
$E[b(Y(\mathrm{Z}))]=p_{Y(\mathrm{Z})}(1)=p_X(1)(1-\epsilon_0)$ respectively.
Therefore, given an energy constraint $B$, the $B$-admissible input $X$ for the multicast information/energy communication
is with a probability of $1$ such that
\begin{equation}\label{eqn:ex-admissible}
p_X(1)\ge\max\left\{\frac{B-\epsilon}{1-2\epsilon}, \frac{B}{1-\epsilon_0}\right\},
\end{equation}
for all
\begin{equation}
0\le B\le B_{\max}=\min\{1-\epsilon, 1-\epsilon_0\}.
\end{equation}

For the point-to-point case, the capacities of BSC and Z channels without input constraints were discovered in \cite{Silverman1955},
which are listed as the follows.
Let $h_{\mathrm{b}}(\cdot)$ be the binary entropy function.
For BSC, the capacity is
\begin{equation}
C_{\mathrm{BSC}}=\max_{p_X}\left\{h_{\mathrm{b}}\big(p_{Y(\mathrm{BSC})}(1)\big)-h_{\mathrm{b}}(\epsilon)\right\},
\end{equation}
where the maximum $1-h_{\mathrm{b}}(\epsilon)$ is achieved when $p_X(1)=0.5$;
for Z-channel, the capacity is
\begin{equation}
C_{\mathrm{Z}}=\max_{p_X}\left\{ h_{\mathrm{b}}\big(p_{Y(\mathrm{Z})}(1)\big)-p_X(1)h_{\mathrm{b}}(\epsilon_0)\right\},
\end{equation}
where the maximum $\log_2 \left(1+2^{-h_{\mathrm{b}}(\epsilon_0)/(1-\epsilon_0)}\right)$ is achieved when
\begin{equation}
p_X(1)=\frac{1}{(1-\epsilon_0)\left(1+2^{h_{\mathrm{b}}(\epsilon_0)/(1-\epsilon_0)}\right)}.
\end{equation}

From \eqref{eqn:nth-cefunction}, the multicast capacity-energy function for these two channels $\Theta$ can be written as
\begin{multline}\label{eqn:ex-capacity}
C_{\mathrm{O}}(B)=\max_{X\mbox{ s.t. } \eqref{eqn:ex-admissible}}\min\{h_{\mathrm{b}}\big(p_{Y(\mathrm{BSC})}(1)\big)-h_{\mathrm{b}}(\epsilon),\\
h_{\mathrm{b}}\big(p_{Y(\mathrm{Z})}(1)\big)-p_X(1)h_{\mathrm{b}}(\epsilon_0)\}.
\end{multline}

Let $X^*(B)$ be the capacity-achieving input in \eqref{eqn:ex-capacity}.
An example with $\epsilon=0.12$ and $\epsilon_0=0.3$ is depicted in Fig.~\ref{fig:ex}.
In Fig.~\ref{fig:ex}, the mutual informations of both channels with input $X^*(B)$ were plotted individually.
Since the multicast capacity is dominated by the worst channel, the achievable region is the intersection of these two curves.
The $C_{\mathrm{O}}(B)$ is simply the minimum between both curves.
\end{comment}

\begin{figure}
  \centering
    \includegraphics[width=3.5in]{ex.png}\vspace{-0.1in}
  \caption{An example of two-receivers multicast communication. Two binary channels were considered: a BSC with crossover probability $0.12$ and
  a Z-channel with a $1$ to $0$ probability $0.3$.}
  \vspace{-0.15in}
  \label{fig:ex}
\end{figure}

\section{Gaussian Channels}
Now consider multicast over $L$ discrete-time, continuous-alphabet, memoryless channels.
Given an input $x$, the received energy at receiver $\ell$ is
\begin{equation}
\rho_\ell(x)=\int Q_\ell(y|x)b_{\ell}(y)dy,
\end{equation}
where $Q_\ell(y|x)$ is the transition probability of the $\ell$th channel.
Let $F(x)$ be the cumulative distribution function (cdf) of $X$;
the expectation of $\rho_\ell(X)$ is:
\begin{IEEEeqnarray}{rCl}
E[\rho_{\ell}(X)]&=&\int\rho_\ell (x)dF(x)\\
&=&\int\int Q_\ell(y|x)b_{\ell}(y)dydF(x)\\
&=&\int\left(\int Q_\ell(y|x)dF(x)\right)b_{\ell}(y)dy\\
&=&E[b_{\ell}(Y(\ell))],
\end{IEEEeqnarray}
which indicates that the output energy constraint can be considered as the function of $\rho_\ell(x)$ at the input side.
Theorem \ref{thm:letterization} can be modified for continuous-alphabet channels to:
\begin{equation}
C(B)=\sup_{\substack{X:E[\rho_{\ell}(X)]\ge B,\\ \forall \ell\in\Theta}}\min_{\ell\in\Theta} I(X;Y(\ell)).
\end{equation}

Consider all $L$ channels having additive white Gaussian noise ${\cal N}(0,\sigma_\ell^2)$
and the energy function $b_{\ell}(y)=y^2$.
Then the received energy at the $\ell$th receiver from input $x$ is
\begin{equation}
\rho_\ell(x)=\int_{-\infty}^{\infty}\frac{y^2}{\sigma_\ell\sqrt{2\pi}}\exp\left\{-\frac{(y-x)^2}{2\sigma_\ell^2}\right\}dy=x^2+\sigma_\ell^2,
\end{equation}
and the expectation is
\begin{equation}
E[\rho_\ell(X)]=E[X^2+\sigma_\ell^2]=\rho_X^2+\sigma_\ell^2,
\end{equation}
where $\rho_X^2\triangleq E[X^2]=\int x^2dF(x)$.
We impose a peak power constraint so the input alphabet is
$\cX=[-P,P]\subset\mathbb{R}$ and then denote the multicast capacity-energy function as $C(B, P)$.

Let $\cF_\ell$ be the space of all cdfs on the finite interval $[-P,P]$ such that, for any $F_\ell\in\cF_\ell$,
\begin{equation}
E_{F_\ell}[\rho_\ell(X)]\triangleq\int_{-P}^P x^2dF_\ell(x)+\sigma_\ell^2 \ge B.
\end{equation}
Since $\cF_\ell\subset\cF_{\ell'}$ if $\sigma_\ell^2\le\sigma_{\ell'}^2$,
the multicast energy constraint \eqref{eqn:energy-constraint} is equivalent to
$
E[\rho_{\ell_{\min}}(X)]\ge B
$,
where $\ell_{\min}=\argmin_{\ell}\sigma_\ell^2$.
Also, since mutual information of a Gaussian channel decreases as the channel's variance increases,
$\min_\ell I(X;Y(\ell))=I(X;Y(\ell_{\max}))$ where $\ell_{\max}=\argmax_{\ell}\sigma_\ell^2$.
Therefore, the multicast capacity-energy function can be rewritten as
\begin{equation}\label{eqn:energy-constraint-gaussian}
C(B,P)=\sup_{X\sim F\in\cF_{\ell_{\min}}}I(X;Y(\ell_{\max})).
\end{equation}

From \cite[Theorem 7]{Varshney2008},
the optimization problem \eqref{eqn:energy-constraint-gaussian} can be solved by the following theorem.
\begin{thm}
There exists a unique capacity-energy achieving input $X_0\sim F_0$
and a constant $\lambda \ge 0$ such that
\begin{IEEEeqnarray}{rCl}
C(B,P)&=&\max_{X\sim F\in\cF_{\ell_{\min}}}I(X;Y(\ell_{\max}))-\lambda J(F)\\
&=&I(X_0;Y(\ell_{\max}))-\lambda J(F_0),
\end{IEEEeqnarray}
where
\begin{equation}
J(F)\triangleq B-\sigma_{\ell_{\min}}^2-\int_{-P}^Px^2dF(x).
\end{equation}
Moreover, a necessary and sufficient condition for $F_0$ to achieve capacity-energy is
\begin{multline}
\int_{-P}^P[i(x;F_0)+\lambda x^2]dF(x)\\\le I(X_0;Y(\ell_{\max}))+\lambda\int_{-P}^{P}x^2dF_0(x)
\end{multline}
for all $F\in\cF_{\ell_{\min}}$, where
\begin{equation}
i(x;F)=\int Q_{\ell_{\max}}(y|x)\log\frac{Q_{\ell_{\max}}(y|x)}{p(y;F)}dy
\end{equation}
is the marginal information density \cite{Smith1971} and $p(y;F)$ is the output density function based on the input distribution $F$.
\end{thm}

\section{Segmentation Problem}
Let us consider the $L$-receivers segmentation problem in Fig.~\ref{fig:multicast_group}.
Here, the receivers are partitioned into $K$ subsets,
denoted as a segmentation $Q=\{\Theta(k)\}_{k=1}^K$ with $\bigcup_{k\in\{1,\cdots,K\}}\Theta(k)=\Theta$,
and the same signal is transmitted to all receivers in the same group.
As discussed in previous sections, the capacity of the $k$th group can be treated as
a multicast system with $|\Theta(k)|$ receivers;
hence \eqref{eqn:capacity} applies to the multicast capacity for each group.

Given an energy constraint vector $\vec{B}$ for all $L$ receivers.
Let the set of all $\vec{B}$-admissible inputs for the $k$th group be $\cF_{\Theta(k)}=\bigcap_{\ell\in\Theta(k)}\cF_1(\ell)$,
then the corresponding capacity is
\begin{equation}
C_{\Theta(k)}=\max_{X\in\cF_{\Theta(k)}}\min_{\ell\in\Theta(k)}I(X;Y(\ell)).
\end{equation}
A trivial performance characterization of a segmentation $Q$ is the worst capacity over all groups:
\begin{equation}\label{eqn:segmantationcapacity}
C_{Q}=\min_{k\in\{1,\ldots,K\}}C_{\Theta(k)}.
\end{equation}
Our objective is to find the segmentation $Q^*$ which maximizes \eqref{eqn:segmantationcapacity}, i.e.,
$Q^*=\argmax_{Q}C_{Q}$.
This is a $k$-partition problem \cite{Chaimovich1993}, solvable by dynamic programming.

A more interesting performance characterization is as follows.
Since the constraint of sending the same signal to the group reduces the capacity
and the capacity is upper-bounded by removing this constraint (see discussion after Theorem \ref{thm:capacityfunction}),
the \emph{segmentation loss} for $k$th group can be defined as follows.
\begin{equation}\label{eqn:segmentationloss}
\Delta_{\Theta(k)}=\left|C_{\Theta(k)}-\min_{m\in\Theta(k)}\max_{W\in\cF_{\Theta(k)}}I(W;Y(m))\right|.
\end{equation}
It is clear that $\Delta_{\Theta(k)}$ is zero when $|\Theta(k)|=1$.
Then the objective is to find the segmentation $Q^*$ which minimizes the maximum $\Delta_{\Theta(k)}$ of all groups, i.e.,
\begin{equation}
Q^*=\argmin_{Q}\max_{k\in\{1,\ldots,K\}}\Delta_{\Theta(k)},
\end{equation}
which is also a minimax quantizer design problem \cite{VarshneyV2014c}.

\begin{figure}
  \centering
    \includegraphics[width=3.5in]{ex-seg.png}\vspace{-0.1in}
  \caption{An example of measurement \eqref{eqn:segmantationcapacity} for the
  three-receivers multicast communication ($L=3$) with two segmentations ($K=2$).
  Three binary channels were considered: a BSC with crossover probability $0.3$ denoted by BSC($0.3$),
  a Z-channel with a $1$ to $0$ probability $0.6$ denoted by Z($0.6$) and
  a Z-channel with a $1$ to $0$ probability $0.65$ denoted by Z($0.65$).}
  \vspace{-0.15in}
  \label{fig:ex-seg}
\end{figure}

\begin{figure}
  \centering
    \includegraphics[width=3.5in]{ex-segloss.png}\vspace{-0.1in}
  \caption{An example of measurement \eqref{eqn:segmentationloss} for the
  same multicast communication in Fig.~\ref{fig:ex-seg}.
  Since $\Delta_\Theta=0$ when $|\Theta|=1$, we only consider the group of size $2$.}
  \vspace{-0.15in}
  \label{fig:ex-segloss}
\end{figure}

An example of clustering three receivers into two groups is given in Figs.~\ref{fig:ex-seg} and \ref{fig:ex-segloss},
in which a BSC($\epsilon=0.3$) and two Z-channels with $\epsilon_0=\{0.6,0.65\}$ were considered.
Fig.~\ref{fig:ex-seg} shows \eqref{eqn:segmantationcapacity} for every segmentation,
indicating that grouping two Z-channels yields the optimal capacity.
However Fig.~\ref{fig:ex-segloss} shows the opposite, when using segmentation loss \eqref{eqn:segmentationloss},
grouping the BSC with any other Z-channel yields zero segmentation loss.
Hence, in this example, the grouping with the optimal capacity is not best when segmentation loss is the main concern.

\section{conclusion}
This paper discussed the simultaneous multicast of energy and information, where the effective channel is equivalent to a constrained compound channel. We studied the capacity for this channel, and derived a single-letter capacity expression under a received energy constraint.
The segmentation problem was also discussed to analyze the situation where receivers are partitioned into related groups. An interesting avenue for future work is to extend the code constructions in \cite{TandonMV2016} and \cite{WuTVM2017} to provide explicit constrained codes for multicast SIET.

\appendices

\section{Unequal Energy Constraints}
\label{append:letterization_general}
To show that the single-letter form can be also derived for unequal energy constraints
$\vec{B}=(B_1,\ldots,B_L)$ and harvest functions $\vec{b}=(b_1,\ldots,b_L)$,
we first prove the domain ${\cal B}$ in \eqref{def:domain} is convex.

\begin{thm}
If $\vec{B}_1=((B_1)_1,(B_1)_2, \ldots, (B_1)_L)$ and $\vec{B}_2=((B_2)_1,(B_2)_2, \ldots, (B_2)_L)$ are both in ${\cal B}$,
then $(\alpha\vec{B}_1+\beta\vec{B}_2)\in{\cal B}$ for all $\alpha\in[0,1]$ and $\beta=1-\alpha$.
\end{thm}
\begin{IEEEproof}
Let $\vec{q}_1=\left((q_1)_1,(q_1)_2,\ldots,(q_1)_{|{\cal X}|}\right)$ and $\vec{q}_2=\left((q_2)_1,(q_2)_2,\ldots,(q_2)_{|{\cal X}|}\right)$ be nonnegative row vectors such that
$\sum_{i=1}^{|{\cal X}|}(q_{j})_i=1$ and
$\vec{q}_j\cdot[P(\ell)]\cdot\vec{b}_\ell=(B_j)_\ell$ for all $j\in\{1,2\}$ and $\ell\in\Theta$.
We have
$(\alpha\vec{q}_1+\beta\vec{q}_2)\cdot[P(\ell)]\cdot\vec{b}_\ell=(B_1)_\ell+(B_2)_\ell$ for all $\ell\in\Theta$.
Since $\sum_{i=1}^{|{\cal X}|}(\alpha(q_1)_i+\beta(q_2)_i)=\alpha+\beta=1$,
we conclude that ${\cal B}$ is convex.
\end{IEEEproof}

The following theorem that the $n$th multicast capacity-energy function for unequal constraints can be evaluated as a single-letter expression in a similar way.

\begin{thm}\label{thm:letterization}
For any DMC, $C_n(\vec{B})=nC_1(\vec{B})$ for all $n\in\mathbb{Z}^+$ and $\vec{B}\in{\cal B}$.
\end{thm}
\begin{IEEEproof}
To prove the equality, we start with the case of ``$\le$''.
Let $X_1^n$ be the channel input with the corresponding outputs $\{Y(\ell)_1^n\}_{\ell\in\Theta}$ which achieves $C_n(\vec{B})$,
then we have $E[b_\ell(Y(\ell)_1^n)]\ge nB_\ell$ for all $\ell\in\Theta$.
We also let $X^*$ be the channel input with the corresponding outputs $\{Y^*(\ell)\}_{\ell\in\Theta}$ such that
$p_{X^*}=\sum_{i=1}^n \frac{1}{n}p_{X_i}$.
Due to the memoryless, We have
\begin{IEEEeqnarray}{rCl}
E[b_\ell(Y^*(\ell))]&=&\frac{1}{n}\sum_{i=1}^nE[b_\ell(Y(\ell)_i)]\\
&=&\frac{1}{n}E[b_\ell(Y(\ell)_1^n)]\ge B_\ell
\end{IEEEeqnarray}
and $I(X_1^n;Y(\ell)_1^n)\le\sum_{i=1}^nI(X_i;Y(\ell)_i)$ for all $\ell\in\Theta$.
Hence
\begin{IEEEeqnarray}{rCl}
C_n(\vec{B})&=&\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n)\\
&\le&\min_\ell \sum_{i=1}^nI(X_i;Y(\ell)_i)\\
&=&n\min_\ell \frac{\sum_{i=1}^nI(X_i;Y(\ell)_i)}{n}\\
&\le&n\min_\ell I(X^*;Y^*(\ell))\\
&\le&nC_1(\vec{B}).
\end{IEEEeqnarray}

For the case of ``$\ge$'',
we let $X$ be the channel input with the distribution $p_X$ and the corresponding outputs $\{Y(\ell)\}_{\ell\in\Theta}$ achieving $C_1(\vec{B})$.
A sequence of channel inputs $X_1^n=(X_1,X_2,\cdots,X_n)$ is created, in which each $X_i$ is i.i.d. to $X$.
Let $\{Y(\ell)_i^n\}_{\ell\in\Theta}$ be the corresponding channel outputs of $X_i^n$.
By definition, $E[b_\ell(Y(\ell)_1^n)]=nE[b_\ell(Y(\ell))]\ge nB_\ell$ for all $\ell\in\Theta$.
Therefore
\begin{IEEEeqnarray}{rCl}
C_n(\vec{B})&\ge&\min_{\ell\in\Theta}I(X_1^n;Y(\ell)_1^n)\\
&=&\min_{\ell\in\Theta}\sum_{i=1}^nI(X;Y(\ell))\\
&=&nC_1(\vec{B}).
\end{IEEEeqnarray}
\end{IEEEproof}

\bibliographystyle{IEEEtran}
\bibliography{abrv,conf_abrv,lrv_lib}

\end{document}

config: arxiv


TITLE: Inequality constraints, probability distributions, and integer partitions
QUESTION [2 upvotes]: I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus,
$a + b + c = 10$
Additionally it has the constraint
$a \geq  b \geq  c$
The allowable triples will correspond to the integer partitions of 10 with width 3, such as:
$(7, 2, 1)$
If you were given a ball and wanted to know the probability that it can from urn $a$, it seems reasonable to sum all the balls in the urn $a$ for all 14 partitions, divided by the total number of balls. In this particular case we would have,
$total \ ball \ count = balls \ per \ partition * partition \ count = 10 * 14 = 140$
$Prob(a) = 90  / 140 \approx .64$
So I have a few questions.
First, am I even on the mark? Is this a valid approach for calculating the probability that the ball comes from urn $a$? If so has this approached been discussed in literature, hopefully in more generality?
I am specifically interested in what the asymptotic behavior is as the number of balls and urns increase.
Here is the code I am using in Haskell to calculate probabilities:
http://hpaste.org/fastcgi/hpaste.fcgi/view?id=27026#a27026

REPLY [3 votes]: Whether this is a valid approach for calculating the probability will depend on what assumptions you are making on the probabilities of the various partitions. In particular, this calculation seems to require for its validity that all partitions be equally likely. Do you actually want (10,0,0) to be as likely as (4,3,3)?
Edit: Concerning asymptotic behavior, take the case where the number of balls equals the number of urns; call it $n$. One can ask for the proportion of balls in the first urn as a function of $n$, which is the same as asking for the average size of the biggest part in a partition of $n$ (which is also the same as asking for the average number of parts in a partition of $n$). This was studied by Kessler and Livingston, The expected number of parts in a partition of $n$, Monatshefte Math 81 (1976) 203-212. Let $P(n)$ be the total number of parts in all the partitions of $n$ (which is the same as the sum of the largest parts in all the partitions of $n$), and let $p(n)$ be the number of partitions of $n$. Then $${P(n)\over p(n)}=\sqrt{3n\over2\pi}(\log n+2\gamma-\log(\pi/2))+O(\log^3n)$$ where $\gamma$ is Euler's constant. More information at sequence A006128 at the Online Encyclopedia of Integer Sequences.
EDIT 15 July: Let $f_{j,k}(n)$ be the sum of the $j$-th biggest parts over all partitions of $n$ into at most $k$ parts, $$f_{j,k}(n)=\sum_{\eqalign{a_1\ge\dots\ge a_k&\ge0\cr a_1+\dots+a_k&=n\cr}}a_j,$$ and let $p_k(n)$ be the number of partitions of $n$ into at most $k$ parts. You are asking for the asymptotic behavior of $f_{j,k}(n)/np_k(n)$ as $k$ and $n$ increase, but it is not clear how you want them to increase (that is, what relation they should bear to each other as they increase), nor is it clear what you want $j$ to be doing while $k$ and $n$ are increasing. Perhaps you are only interested in the case $j=1$, which is the case in your example.
In any event, you need asymptotics for $p_k(n)$. From the generating function $$\sum_np_k(n)x^n={1\over(1-x)(1-x^2)\cdots(1-x^k)}$$ one can get, for fixed $k$, $p_k(n)=C_kn^{k-1}+O(n^{k-2})$, where $C_k=(k!(k-1)!)^{-1}$ (I hope someone checks my calculations here).
Now I can tell you something about $f_{k,k}(n)$ - this is the case when you're interested in the $\it last$ urn. Let's write $g_k(n)=f_{k,k}(n)$. It's not hard to prove that $g_k(n+k)-g_k(n)=p_k(n)$, from which it follows that $$\sum_ng_k(n)x^n={x^k\over(1-x)\cdots(1-x^{k-1})(1-x^k)^2}$$ and then you get, again for fixed $k$, $g_k(n)=D_kn^k+O(n^{k-1})$, where $D_k=k^{-1}(k!)^{-2}$. So, the probability that the ball is from the last urn approaches $k^{-2}$ as $n$ increases with $k$ fixed.
I didn't do so well with the first urn. Let's just take the case $k=3$. We want $$h_3(n)=f_{1,3}(n)=\sum_{\eqalign{a\ge b\ge c&\ge0\cr a+b+c&=n\cr}}a.$$ Starting with $n=1$, this produces the sequence $1,3,6,11,17,27,37,52,69,90,113,144\dots$. I didn't find this sequence in the Online Encyclopedia of Integer Sequences. I think, but can't prove, that the generating function is $$\sum_nh_3(n)x^n={x+3x^2+4x^3+3x^4\over(1-x)^2(1-x^3)^2}.$$ This gives $h_3(n)=(11/216)n^3+O(n^2)$. Together with $p_3(n)=(1/12)n^2+O(n)$, you get the probability of the ball coming from the first of the three urns converging to 11/18, as $n$ increases (and conditional on my unproved formula for the generating function).
EDIT 8 February 2011: There's a typo in the last display, it should be $$\sum_nh_3(n)x^n={x+3x^2+4x^3+3x^4\over(1-x^2)^2(1-x^3)^2}.$$ George Andrews wrote out a proof of this formula (and his method should apply more generally to $\sum_nf_{j,k}(n)x^n$). Let $a=r+s+t$, $b=s+t$, $c=t$, then the generating function is $\sum_{r,s,t}(r+s+t)x^{r+2s+3t}$, which can be split up into three pieces each of which is two geometric progressions times a sum of the shape $\sum_uuy^u$. That last sum is well-known, and after some algebra the conjectured formula pops out.


text:
\section{Floor Function/Examples/Floor of Binary Logarithm of 35}
Tags: Examples of Floor Function

\begin{theorem}
:$\floor {\lg 35} = 5$
where:
: $\lg x$ denotes the [[Definition:Binary Logarithm|binary logarithm ($\log_2$)]] of $x$
\end{theorem}

\begin{proof}
{{begin-eqn}}
{{eqn | l = \lg 32
      | o = \le
      | m = \lg 35
      | mo= <
      | r = \lg 64
      | c =
}}
{{eqn | ll= \leadsto
      | l = 5
      | o = \le
      | m = \lg 35
      | mo= <
      | r = 6
      | c =
}}
{{end-eqn}}
Hence $5$ is the [[Definition:Floor Function|floor]] of $\lg 35$ by definition.
{{qed}}
\end{proof}

config: wiki


TITLE: Extend subset of a ring to a ring
QUESTION [1 upvotes]: I have the following:
Let A be a ring and B $\subset$ A a subring of A, and X $\subset$ A a subset of A. Show that exists a subring B[X] of A such that X $\subset$ B[X] and B $\subset$ B[X] and any other subring of A that contains B and X, contains B[X]
I'm not sure if what I'm thinking is right, but I wanted to extend X to a ring X', and then define B[X]:={ bx$\in$ A / b$\in$B and x$\in$X' }
X' should be defined as the set generated by all of the elements of X. Is this right?

REPLY [1 votes]: In this answer, I am making the assumption that $A$ is unital; that is, that there is an element $1_A \in A$ such that $1_A a = a1_A = a$ for all $a \in A$.  This assumption is, I believe, in accord with current the current usage of the word "ring", the recently coined term "rng" being reserved for entities satisfying all the ring axioms except the existence of a multiplicative unit; so, for example, $\Bbb Z$ is a ring, but $2\Bbb Z$ is a rng.  And of course, since $B \subset A$ is a subring, we assume it inherits the multitplicative identity of $A$ so that $1_A$ is the unit element of $B$.  I'm pretty sure, though I have not worked through the details, that this result is correct even in the even $B$ is merely a rng, but I believe the demonstration in that case to be somewhat more delicate; so here, I'll stick with unital rings.
Having said these things,
The way I would attack this problem is to consider the set $S$ defined by
$S = \{ \sum_1^n \prod_1^m y_j \mid m, n \in \Bbb N, y_j \ \in B \cup X \}; \tag 1$
that is, $S$ is the set of all finite sums of finite products of elements selected from $B$ and $X$.  It is clear that $B \subset S$ and $X \subset S$; we show that in fact $S$ is a subring of $A$.  Since $0 \in B$, it is clear that $0 \in S$; furthermore, since the sum of any two sums of finite products of elements from $B$ and $X$ is again a sum of such products, it is also clear that $S$ is closed under the "$+$" operation of $A$.  The additive inverse of any such sum $s$ may be had by multiplying it by $-1_A$, which we have assumed is also an element of $B$.  Thus, since the other properties of ring addition are inherited from $A$, $S$ forms an additive subgroup of $A$.  Furthermore, it is easy to see that the product of two elements of $S$ may again be expressed as a finite sum of finite products of elements taken from $B$ and $X$; and again, since axiomatic properties of the multiplication operation in $A$ will be inherited by elements of $S$, it ($S$) will obey all the usual ring axioms for multiplication, including the distributive axioms and so forth.  So $S$ is indeed a subring of $A$.
If $R$ is a subring of $A$ such that $B \subset R$ and $X \subset R$, and $m, n \in \Bbb N$ and $y_j \in B \cup X$, it is easy to see that for $s \in S$,
$s = \sum_1^n \prod_1^m y_j \in R, \tag 2$
since (as has in fact been made clear already in the above), $s$ is formed from the $y_i$ via a finite number of the ring operations $+$ and $\cdot$; thus $S \subset R$ for every subring $R$ containing $B$ and $X$.  In fact we have
$S \subset \bigcap_{B, X \subset R \subset A; \; R \; \text{a subring}}R, \tag 3$
and since $S$ is a subring satisfying $B, X \subset R \subset A$, we see that it actually occurs on the right of (3), whence in fact
$S = \bigcap_{B, X \subset R \subset A; \; R \; \text{a subring}}R. \tag 4$
(4) shows that every subring R of $A$ with $B, X \subset R$ contains $S$; thus we may take
$S = B[X]. \tag 5$
There is no need to extend $X$ to a subring $X'$; the process of taking finite sums of finite products accomplishes this extension, and in any event would be "executed" in so extending $X$.  In any event, one would need to take sums of the products $bx \in A$ occurring in the OP's definition of $B[X]$ since in general, the set of such products will not be additively closed in and of itself.  Finally, defining $X'$ to be the set of elements generated from $X$, for example via the finite sums of finite products procedure, may not produce a subring; it is not clear that we will necessarily have a multiplicative unit in $X'$; perhaps $X'$ will be a rng, but I leave that discovery to my readers.
It is worth noting that the finite sums of finite products construction is similar to taking the product of a set of ideals; the idea in each case is to obtain an object closed under addition etc.


TITLE: Continuous curve interpolating a list of points
QUESTION [1 upvotes]: I need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way.
Found out that what I need is a spline, however cannot find one that behaves how I need. $n$-degree Bezier curves and B-splines don't pass through the $n$ points, just move from the first to the last one using the others as control points.
A Bezier spline of $n-1$ Bezier curves (defined between each pair of adjacent points) should do the trick, but to have a smooth result I would need some control points which I don't have and don't know how to compute.
Any hints?

REPLY [1 votes]: A parametric cubic spline is what you want; I've talked about them at some length in this answer (and in probably a few other places on this site, since it seems to be a common question), so I won't repeat myself now. Suffice it to say that yes, Bézier isn't what you need here.


TITLE: Can we find a regular cover for the set of integers with the following conditions?
QUESTION [0 upvotes]: Can we find a regular cover for the set of integers with a finite collection of congruence wherein every integer belong to at least one of the classes in the system and the system of congruence consists only of odd prime modulus, each modulus is repeated exactly twice and is greater or equal to 7?  Can I say that it is not possible since if a prime p divides one of the modulus, then it should divide at least p of them?

REPLY [0 votes]: Since the moduli $p_i$ are all at least $7,$ and we have used only two residue classes in the proposed covering for each modulus, we can choose a third residue class $c_i$ for each  mod $p_i.$ Then one can use Chinese remainder to get an $x$ with $x \equiv c_i$ mod $p_i$ for each of the (finitely many) $i$. This $x$ will not satisfy any of our proposed congruences in the cover.
Note the same argument works for any system of moduli which are pairwise coprime and all at least $3.$


TITLE: Can a space with $\pi_1(X)=F_2$ have a nontrivial covering space which is homeomorphic to $X$?
QUESTION [11 upvotes]: Any finite sheeted connected cover of the circle is again homeomorphic to a circle. On the group level, this is consistent with the fact that subgroups of $\mathbb Z$ are isomorphic to $\mathbb Z$. This leads to the question of whether one could find nontrivial covers which are still homeomorphic to the base space for other groups that have subgroups isomorphic to themselves. In particular, consider the free group on 2 generators $F_2$. It has plenty of subgroups isomorphic to $F_2$, so one can ask
Question Is there a space $X$ with $\pi_1(X)\cong F_2$ which has a nontrivial cover $p\colon E\to X$, where $E\cong X$?
Clearly the standard example of $X=S^1\vee S^1$ doesn't work, but perhaps starting with $X$ as a graph of infinite valence might work, or perhaps there is an even more clever construction.

REPLY [6 votes]: Let $G\subset F_2=\langle a,b\rangle$ be the subgroup generated by $a^2$ and $b^2$.  Let $X_0=S^1\vee S^1$ and recursively define $X_{n+1}$ to be the covering space of $X_n$ associated to $G$, identifying $\pi_1(X_n)\cong G$ with $F_2$.  Note that there is an embedding $X_0\to X_1$ that induces an isomorphism on $\pi_1$, mapping the two loops of $X_0$ to the loops in $X_1$ corresponding to $a^2$ and $b^2$.  Taking the covering spaces associated to $G$ on both sides, this induces an embedding $X_1\to X_2$, and then $X_2\to X_3$, and so on.  Let $X$ be the colimit of this sequence of embeddings $X_0\to X_1\to X_2\to\dots$.
Since each of the maps $X_n\to X_{n+1}$ is a closed embedding, $\pi_1(X)$ is the colimit of the groups $\pi_1(X_n)$ and thus is isomorphic to $F_2$.  Now, the covering maps $p_n:X_{n+1}\to X_n$ are compatible with the inclusion maps in the colimit, and thus induce a map $p:X\to X$ which induces the inclusion $F_2\cong G\to F_2$ on $\pi_1$.  So, if this colimit map $p$ is still a covering map, it is the desired example.
To sketch a proof that $p$ is a covering map, fix a pair of points $x,y\in X_0$, one in each circle, and neither one a dyadic rational fraction of the way around the circle.  This means that neither $x$ nor $y$ will ever become a vertex in the natural CW-complex structure on any $X_n$.  Then for all $n$, $X_n\setminus\{x,y\}$ is evenly covered by $p_n$ (identifying $x$ and $y$ as points of $X_n$ via the embedding $X_0\to X_n$).  It follows then that $X\setminus\{x,y\}$ is evenly covered by $p$.  Varying $x$ and $y$, sets of the form $X\setminus\{x,y\}$ cover $X$.
(Intuitively, $X$ looks like $S^1\vee S^1$ with infinitely dense tree-like "fuzz" attached at each dyadic rational along each circle, where the fuzz itself has fuzz attached at each dyadic rational along each edge, and then the fuzz's fuzz also has fuzz, and so on.)

REPLY [5 votes]: So I think the answer is simple. Start with $S^1\vee S^1$ and take any cover for which $\pi_1$ of that cover is a free group of rank 2. Now thicken these examples slightly by replacing each vertex by an open disk and each edge by an open strip. So $X$ is homeomorphic to an open disk with two closed disks removed. The cover of $X$ will also include thickenings of the tree parts of the graph, but because we are not including the boundary, these will just be homeomorphic to open disks glued to the rest of the graph.


TITLE: Nice result using Bernoulli variables?
QUESTION [0 upvotes]: I'm preparing an oral exam concerning independant Bernoulli variables sequences. I have to present two 'big' results concerning those, and each result must take more or less 15' of demonstration.
Problem is I'm not so interested in probabilities, so I'd like to find results that are also used in other areas (algebra preferably!). For instance, my first demonstration will concern Bernstein's polynomials.
Any idea ?

REPLY [0 votes]: I don't know if it's "big" enough, but one neat trick is generating a random variable with a Cantor distribution (i.e. its CDF is the Cantor "devil's staircase", and it is continuous but singular):
$$X = 2 \sum_{j=1}^\infty 3^{-j} B_j $$
where $B_j$ are independent Bernoulli$-1/2$ random variables.
You even get to bring in another Bernoulli, if you compute the cumulants of this distribution.
But if you want something related to combinatorics, consider
$$S_n = \sum_{j=1}^n k B_k$$
Its probability mass function
$$ \mathbb P(S_n = m) =  \dfrac{f_n(m)}{2^n}$$ where
$f_n(m)$ is the number of partitions of $m$ into distinct parts that are all $\le n$.


TITLE: Let $E$ be a subset of $\textbf{R}^{n}$. Then $E$ is compact iff it is closed and bounded.
QUESTION [0 upvotes]: Let $(\textbf{R}^{n},d)$ be a Euclidean space with either the Euclidean metric, the taxicab metric, or the sup norm metric. Let $E$ be a subset of $\textbf{R}^{n}$. Then $E$ is compact iff it is closed and bounded.
MY ATTEMPT
Since such metrics are equivalent, it suffices to consider one of them.
Once every compact set is closed and bounded within a metric space, we have only to prove the converse implication.
Since $E\subseteq\textbf{R}^{n}$ is bounded, we conclude that $E\subseteq B(0,r)$ for some $r > 0$. More precisely,
\begin{align*}
|x^{(k)}_{j}| \leq |x^{(k)}_{1}| + |x^{(k)}_{2}| + \ldots + |x^{(k)}_{n}| < r
\end{align*}
In particular, for any sequence $(x^{(k)})_{k=m}^{\infty}\in E$, we conclude each coordinate sequence $(x^{(k)}_{j})_{k=m}^{\infty}$ is bounded.
Therefore, according to the theorem of Bolzano-Weierstrass, each sequence $(x^{(k)}_{j})_{k=m}^{\infty}$ admits a subsequence  which converges to $x_{j}$.
Once a sequence converges iff each coordinate converges, $(x^{(k)})_{k=m}^{\infty}$ has a subsequence which converges to $x = (x_{1},x_{2},\ldots,x_{n})$.
But $E$ is also closed. This means that any sequence of points $x_{k}\in E$ which converges to $x$ implies that $x\in E$.
Gathering all these results, we conclude that if $E$ closed and bounded, then every sequence $x_{n}\in E$ admits a convergent subsequence.
That is to say, $E$ is compact, and we are done.
Could someone critique my solution?

REPLY [2 votes]: There is one problem in your solution. You can't just pick a random convergent subsequence of $(x_j^{(k)})$ for every $j$. That way you will get $n$ subsequences which have nothing to do with each other, the indexes might be different. What are the indexes of the subsequence of $(x^{(k)})$ then?
Let's assume $n=2$ to make the writing less complicated, the same proof works for every $n$. What you need to do is the following: first pick a convergent subsequence of $(x_1^{(k)})$, let's call it $(x_1^{(k_j)})_{j=1}^\infty$, and let's call the limit $x_1$. Next, look at the sequence $(x_2^{(k_j)})_{j=1}^\infty$. It has a convergent subsequence $(x_2^{(k_{j_L})})_{L=1}^\infty$ which converges to some $x_2$. Now, since a subsequence of a convergent sequence converges to the same limit we also have $(x_1^{(k_{j_L})})\to x_1$ when $L\to\infty$. And now we can indeed say that:
$(x_1^{(k_{j_L})}, x_2^{(k_{j_L})})\to (x_1,x_2)$ as $L\to\infty$.
So now $(x^{(k_{j_L})})_{L=1}^\infty$ is indeed a convergent subsequence of $(x^{(k)})$.


TITLE: Special function or Weierstrass?
QUESTION [0 upvotes]: Given that $f(x)$ is integrable on any $[0,a]$, such that $\lim_{x\to+\infty} f(x)=1$

Prove that: $$\lim_{t\to 0+}\int_0^{\infty}t\text{e}^{-tx}f(x)\text{d}x=1$$

REPLY [3 votes]: The function $g:z\mapsto\sup\limits_{x\geqslant z}|f(x)-1|$ is finite everywhere and, for every $z$,
$$
\left|\int_0^{+\infty}t\mathrm e^{-tx}f(x)\mathrm dx-1\right|\leqslant(1-\mathrm e^{-tz})\cdot g(0)+\mathrm e^{-tz}\cdot g(z)\leqslant t\cdot z\cdot g(0)+ g(z)
$$
Thus, for every $z$,
$$
\limsup\limits_{t\to0}\ \left|\int_0^{+\infty}t\mathrm e^{-tx}f(x)\mathrm dx-1\right|\leqslant g(z).
$$
Since $g$ has limit $0$ at $+\infty$, the proof is complete.


TITLE: Arc given two points and their start / end tangents
QUESTION [0 upvotes]: Given two points ($p_0$ and $p_1$) and their (start/end) tangents ($t_0$ and $t_1$), how can I find the arc that joins them?
Thanks!

REPLY [2 votes]: If you are looking for a circular arc, all you need to do is to draw the two lines perpendicular to each tangent at their points $P_0$ and $P_1$: the point of intersection of these lines is the center $C$ of the circle. Unfortunately that works only if the distance of $C$ from $P_0$ and $P_1$ is the same, but in general this is not guaranteed to be true and no fitting circular arc exists.