text:
\section{Convergence of kernel operator in finite data approximation}\label{s:Particle}

We now turn to the ``variance'' error, i.e. the convergence of the finite data approximation as the sample size $M \to \infty$. In this section we begin by showing the convergence of the discretised Gaussian kernel $\D^M_\eps$ to the continuum limit $\D_{\eps}$. We prove convergence first pointwise for fixed functions, then extend to convergence in norm on fixed functions, and then
finally to norm-convergence of operators.

Recall from \RR{eq:DMeps}{eq:Deps} that we defined the operators $\D_\eps$ and $\D^M_\eps$ as
\begin{equation} (\D_\eps \phi)(x)= \int g_{\eps,L}(x-y) \phi(y) \rho(y) \,\dd y \label{eq:DepsAgain} \end{equation}
and
\begin{equation} (\D^M_\eps \phi)(x) = \frac{1}{M} \sum_{i=1}^M g_{\eps,L}(x-x^i) \phi(x^i). \label{eq:DMepsAgain} \end{equation}
Since the $\{x_i\}$ are sampled from the measure $\rho$, the continuous operator $\D_\eps \phi$ is the expectation of the discretised operator $\D^M_\eps \phi$ with respect to this sampling. Because the discretised operator is the sum of independent random variables $g_{\eps,L}(x-x^i) \phi(x^i)$, it is therefore natural to try to construct central limit theorems.

The basic result we will use for this purpose is the following Bernstein inequality that provides strong quantitative control on the tail probabilities:
\begin{proposition}\label{p:CLTbounds}
	Consider an {\em i.i.d.} collection of bounded, centred random variables $X_i$. Then if $\E[X^2] \leq \nu$ and $c \leq 6 \nu / \| X\|_0$,
	\[ \Pr\left(\left| \frac{1}{M} \sum_{i=1}^M X_i \right| > c \right) \leq 2 e^{-M c^2/6\nu}. \]
\end{proposition}

To deal with the fact that we are using the periodised Gaussian kernel $g_{\eps,L}$ rather than the standard one $g_\eps$, we will require the following proposition:
\begin{proposition}\label{p:GaussianL}
	Define the increasing functions of $\eps$
		\begin{align*} \gamma_{\eps,L} &= \sum_{j \in \mathbb{Z}} e^{-j^2 L^2/2\eps},\\
		\gamma_{\eps,L}' &= \sum_{j=1}^\infty (2j+1) L \eps^{-1/2}  e^{-(2j-1)^2 L^2/8\eps}.
		\end{align*}
	Then for all $x \in [-L/2,L/2]^d$,
	\begin{align*}
		g_{\eps,L}(x) &\leq (1+\gamma_{\eps,L})^d g_\eps(x), \\
		 \sup g_{\eps,L} &\leq \gamma_{\eps,L}^d g_\eps(0), \\
		\Lip g_{\eps,L} &\leq \gamma'_{L,\eps,d} \Lip g_\eps,
	\end{align*}
	where
	\[\gamma'_{L,\eps,d} := e^{-1} + d\gamma_{\eps,L}' \gamma_{\eps,L}^{d-1}.\]
\end{proposition}

The next lemma, on pointwise evaluation of the operators we are interested in, follows from Proposition \ref{p:CLTbounds}.
\begin{lemma}\label{l:Pointwise}
	For all $\phi \in C^0$, $c \leq 3\rhonm$ and $x \in \domain$,
	\[ \Pr\left(| (\D^M_\eps \phi)(x) - (\D_\eps \phi)(x)| > c \phinm\right) \leq 2\exp\left\{-\frac{M c^2}{6 \gamma_{\eps,L}^d (2\pi\eps)^{-d/2} \rhonm}\right\}. \]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{l:Pointwise}]
	Equations \RR{eq:DepsAgain}{eq:DMepsAgain} and the independent sampling of the $x^i$ from $\rho$ mean that $(\D^M_\eps \phi)(x)$ is a sum of {\em i.i.d.} centred, bounded random variables:
	\[ (\D^M_\eps \phi)(x) = \frac{1}{M} \sum_{i=1}^M \mathfrak{g}_x(x^i),\]
	where
	\[ \mathfrak{g}_x(y) = g_{\eps,L}(x-y)\phi(x^i) - \E_{y}[g_{\eps,L}(x-y) \phi(y)]. \]
	The sup-norm of this function is bounded as
	\[ \|\mathfrak{g}_x\|_0 \leq 2\|g_{\eps,L}(x-\cdot)\phi(\cdot)\|_0 \leq 2 (2\pi\eps)^{-d/2} \gamma_{\eps,L}^d \phinm, \]
	and the $L^2$ norm as
	\begin{align*} \E[\mathfrak{g}_x^2] &\leq \E_y[g_{\eps,L}(x-y)^2 \phi(y)^2]\\
	&= \int g_{\eps,L}(x-y)^2 \phi(y)^2 \rho(y) \dd y\\
	&\leq (2\pi\eps)^{-d/2} \gamma_{\eps,L}^d \phinm^2 \rhonm.
	\end{align*}
	From an application of Proposition \ref{p:CLTbounds} the result then follows.
\end{proof}

By using the compactness of our domain $\domain$ we can extend this to bounds on the function norms:
\begin{lemma}\label{l:FunctionNorm}
	There exist constants ${{C}}_{41}, {{C}}_{42}$ depending only on $L,d,\rhonm,\eps_0$ such that for all $\eps < \eps_0$, $\phi \in C^0$ and $c < 3 \rhonm$,
	\[ \Pr\left(\left\| (\D^M_\eps - \D_\eps) \phi\right\|_{0} > 2c \phinm\right) \leq 2 {{C}}_{42} c^{-d} \eps^{-d(d+1)/2} \exp\left\{-{{C}}_{41} M \eps^{d/2} c^2\right\}. \]
\end{lemma}
\begin{proof}[Proof of Lemma \ref{l:FunctionNorm}]
		Firstly, we have the deterministic bound that
		\begin{equation} \Lip (\D_\eps - \D^M_\eps) \phi \leq \Lip \D_\eps \phi + \Lip \D^M_\eps \phi \leq 2\Lip g_\eps \phinm = 2\eps^{-1/2} (2\pi\eps)^{-d/2} \gamma'_{L,\eps,d} \phinm. \label{eq:LipBound}\end{equation}

		Now, define the finite subset of the domain $\domain = [0,L]^d$
		\[ S_\xi = \{ (\xi n_1,\ldots \xi n_d) : n_1, \ldots n_d = 0, \ldots, \lceil L/\xi \rceil-1. \}. \]
		No point in $\domain$ is more than $\sqrt{d} \xi$ away from an element of $S_\xi$, and $S_\xi$ contains no more than $(L/\xi+1)^d$ points.

		By applying Lemma \ref{l:Pointwise} and a union bound, we obtain that for all $x \in S_\xi$
		\[ \Pr\left(\sup_{x\in S_\xi} | (\D^M_\eps \phi)(x) - (\D_\eps \phi)(x)| > c \phinm \right) \leq 2(L/\xi+1)^d \exp\left\{-{{C}}_{41} M \eps c^2\right\}, \]
		where the constant
		\[ {{C}}_{41} := \tfrac{1}{6} \gamma_{\eps_0,L}^{-d} (2\pi)^{d/2} \rhonm^{-1}. \]

	Using the Lipschitz bound \R{eq:LipBound} we can then say that
			\begin{align*} &\Pr\left(\sup_{x\in \domain} | (\D^M_\eps \phi)(x) - (\D_\eps \phi)(x)| > (c +  2\eps^{-1/2} (2\pi\eps)^{-d/2} \sqrt{d} \gamma'_{L,\eps,d} \xi )\phinm \right) \\
			&\qquad \leq 2(L/\xi+1)^d \exp\left\{-{{C}}_{41} M \eps^{d/2} c^2\right\}. \end{align*}
		Setting
		\[\xi = \frac{c \eps^{1/2} (2\pi\eps)^{d/2} \gamma'_{L,\eps,d}}{ 2\sqrt{d}}\]
		we obtain
					\[ \Pr\left(\| (\D^M_\eps-\D_\eps) \phi\|_{0} > 2c \phinm \right) \leq 2\left( \frac{2L\sqrt{d} \eps^{-1/2} (2\pi\eps)^{-d/2}}{c\gamma'_{L,\eps,d}} + 1 \right)^d \exp\left\{-{{C}}_{41} M \eps^{d/2} c^2\right\}, \]
					which requiring that $\eps \leq \eps_0$ and setting
					\[ {{C}}_{42} = \left(2 L \sqrt{d} (2\pi)^{-d/2}/\gamma'_{L,\eps_0,d} + \eps_0^{(d+1)/2} 3 \rhonm\right)^d \]
					gives the required bound.

\end{proof}

We would now like to extend this result to convergence as operators. Recall that we defined for $\zeta > 0$ the complex domains
\[ \domain_\zeta = \{ x + iz \mid x \in \domain, z \in [-\zeta,\zeta]^d \}, \]
so that $\domain \subset \domain_\zeta \subset (\mathbb{C} / L\mathbb{Z})^d$; we also defined the Hardy spaces
\[ H^\infty(\domain_\zeta) = \{ \phi \in C^0(\domain_\zeta) : \phi \textrm{ analytic on }\intr \domain_\zeta\} \]
with $\| \cdot \|_{\zeta}$ being the $C^0(\domain_\zeta)$ norm. In Theorems \ref{t:Delta} and \ref{t:Operator} (presented in Section \ref{s:Theorems}) we show that when the size of $\zeta$ scales with the kernel bandwidth $\sqrt{\eps}$, $\D^M_\eps$ converges in operator norm to $\D_\eps$.

To extend from function-wise convergence to uniform convergence across all functions, we will again make use of a compactness argument: this time, the compact embedding of $H^\infty(\domain_\zeta)$ in $C^0(\domain)$. This choice allows us to obtain good operator convergence bounds in the strong space $H^\infty(\domain_\zeta)$ as, Gaussian convolution $\C_\eps$ maps the weak space $C^0(\domain)$ into the strong space $H^\infty(\domain_\zeta)$ with an $\bigO(1)$ penalty in norm, provided that $\zeta$ is $\bigO(\eps^{1/2})$ (see Proposition \ref{p:WeaktoStrong}).

However, this scaling restriction on $\zeta$, which arises from the width of the Gaussian kernel, leads to a complication. Because the larger complex domain $\domain_\zeta$ is only a relatively small extension of the real domain $\domain$, the number of $C^0(\domain)$ balls required for a covering of $H^\infty(\domain_\zeta)$ is exponentially large in $\eps^{-1/2}$. This jeopardises the Central Limit Theorem bounds obtained in Lemma \ref{l:FunctionNorm}.

However, we can use the Gaussian kernel's localisation to our advantage, as the values of $\D^M_\eps\phi(x), \D_\eps\phi(x)$ more or less depend only on values of $\phi$ inside a ball slightly larger than $\bigO(\eps^{1/2})$. We thus divide our domain $\domain$ up into small, overlapping cubes $\mathcal{E}$ of this size: the complex $\zeta$-fattening $\mathcal{E}_\zeta$ is a sufficiently large extension of $\mathcal{E}$ and on each of these cubes we therefore have acceptable covering numbers.

We will make use of the following quantitative compactness result, proved in Appendix \ref{a:CompactnessProof}. \rii{Note that the analyticity of the Gaussian kernel is crucial for this result.}
\begin{proposition}\label{p:Compactness}
	Let $\mathcal{E} \subset \domain$ be a hypercube of side length $2\ell \geq 2\zeta/\eta_0 $ and, $\mathcal{E}_\zeta$ the closed $\zeta$-fattening of $\mathcal{E}$
	\[ \mathcal{E}_\zeta = \{ x \in \domain_\zeta : d(x,\mathcal{E}) \leq \zeta \}. \]

	There exist constants ${{C}}_{43}, {{C}}_{44}$ dependent only on $\eta_0, d$ such that for each $\xi\in(0,\half)$ there exists a set $\mathcal{S}^{\ell,\zeta}_\xi$ such that for every function $\phi \in H^\infty(\mathcal{E}_\zeta)$ with $\|\phi\|_{H^\infty(\mathcal{E}_\zeta)} \leq 1$,
	\begin{equation} \sup_{\psi \in \mathcal{S}^{\ell,\zeta}_\xi} \|\phi - \psi\|_{C^0(\mathcal{E})} \leq \xi, \label{eq:HardyCovering}\end{equation}
	and the cardinality of $\mathcal{S}^{\ell,\zeta}_\xi$ is bounded by
	\[ |\mathcal{S}^{\ell,\zeta}_\xi| \leq e^{({{C}}_{43} \log \xi^{-1} + {{C}}_{44} \log (\zeta^{-1}\ell))(\zeta^{-1} \ell \log \xi^{-1})^{d}}. \]
\end{proposition}

Using Proposition \ref{p:Compactness} we can prove central limit theorem-style bounds on the operator norm of $\D^M_\eps - \D_\eps$ from a strong space associated with a larger cube $\mathcal{E}$ to a weak space associated with a smaller cube $E$.
\begin{proposition}\label{p:LInfHeads}
	Let $\mathcal{E}$ be as in Proposition \ref{p:Compactness} and let $E$ be a hypercube of side length $2l < 2\ell$ centred inside $\mathcal{E}$. Then there exist positive constants ${{C}}_{44}, {{C}}_{45}, {{C}}_{46}$ dependent only on $\rhonm, d, L, \eta_0, \eps_0$ such that for $c \leq 3 \rhonm$,
	\begin{align*}&\Pr\left(\|(\D^M_\eps - \D_\eps) \ine \|_{H^\infty(\mathcal{E}_\zeta) \to C^0(E)} \geq 3c\right)\\
	&\qquad \leq \exp\left\{({{C}}_{45} \log \eps^{-1} + {{C}}_{46} \log c^{-1} + {{C}}_{44} \log (\ell/\zeta))^{d+1} (\ell/\zeta)^d - \frac{M c^2}{4 (2\pi\eps)^{-d/2} \rhonm}\right\}, \end{align*}
	where $\mathbb{1}_\mathcal{E}$ is the characteristic function of $\mathcal{E}$, considered as a multiplication operator.
	\end{proposition}
	\begin{proof}[Proof of Proposition \ref{p:LInfHeads}]
		The proof proceeds analogously to the proof of Lemma \ref{l:FunctionNorm}.

		Let $\mathcal{S}^{\ell,\zeta}_\xi$ be as in Proposition \ref{p:Compactness}. The difference between the operators can be bounded deterministically by
		 \[ \|\D^M_\eps - \D_\eps\|_0 \leq \|\D^M_\eps\|_0 + \|\D_\eps\|_0 \leq 2 \gamma_{\eps,L}^d (2\pi\eps)^{-d/2}  \]
		 and so, using \R{eq:HardyCovering},
		 \begin{equation} \sup_{\psi \in \mathcal{S}^{\ell,\zeta}_\xi} \left\| (\D^M_\eps - \D_\eps)\ine (\|\phi\|_\zeta^{-1} \phi - \psi)\right\|_{C^0(E)} \leq 2\gamma_{\eps,L}^d (2\pi\eps)^{-d/2} \xi \label{eq:HardyCoverDiff}\end{equation}
		 for all $\phi$ in the unit ball of $H^\infty(\mathcal{E}_\zeta)$.

		 On the other hand, we can apply Lemma \ref{l:FunctionNorm} and a union bound to show that
		  \begin{equation}  \Pr\left(\sup_{\psi \in \mathcal{S}^{\ell,\zeta}_\xi} \| (\D^M_\eps - \D_\eps) \ine \psi \|_{0} > 2c\right) \leq |\mathcal{S}^{\ell,\zeta}_\xi| {{C}}_{42} c^{-d} \eps^{-d(d+1)/2} \exp\left\{-{{C}}_{41} M \eps^{d/2} c^2\right\}. \label{eq:HardyCoverProb}\end{equation}

		 By combining \RR{eq:HardyCoverDiff}{eq:HardyCoverProb} and setting $\xi = (2\pi\eps)^{d/2} c/2\gamma_{\eps,L}^d$ we obtain that
		 \begin{align*}  &\Pr\left(\sup_{\|\phi\|_{H^\infty(\mathcal{E}_\zeta)} \leq 1} \| (\D^M_\eps - \D_\eps) \ine \phi \|_{0} > 3c\right) \\ &\qquad\leq |\mathcal{S}^{\ell,\zeta}_{(2\pi\eps)^{d/2} c/2\gamma_{\eps,L}^d}| {{C}}_{42} c^{-d} \eps^{-d(d+1)/2} \exp\left\{-{{C}}_{41} M \eps^{d/2} c^2\right\},
		 \end{align*}
		 which using that $\eps \leq \eps_0$, $\ell/\zeta \geq \eta_0$ and $c \leq 3 \rhonm$ and the bound on $\mathcal{S}^{\ell,\zeta}_{\xi}$ in Proposition \ref{p:Compactness} gives the required bound.
	\end{proof}

We can also make a deterministic bound on the error that this restriction to the larger cube $\mathcal{E}$ introduces relative to the full diffusion. \rii{In this proposition the Gaussian kernel's exponential decay is crucial.}
\begin{proposition}\label{p:LInfTails}

Let $\mathcal{E}, E$ be as in Proposition \ref{p:LInfHeads}. Then
	\[ \| (\D^M_\eps - \D_\eps) \notine \|_{H^\infty(\domain_\zeta) \to C^0(E)} \leq 2 (1+\gamma_{\eps,L})^d (2\pi\eps)^{-d/2} e^{-(\ell-l)^2/2\eps}. \]
\end{proposition}
\begin{proof}[Proof of Proposition \ref{p:LInfTails}]
	For $x \in E$,
	\begin{align*} |\D^M_\eps \notine \phi (x)| &\leq \frac{1}{M} \sum_{i=1}^M |g(x-x^i)| \notine(x^i) \phinm\\
	&\leq \sup_{y \in \domain\backslash\mathcal{E}} g(x-y) \phinm\\
	&\leq (2\pi\eps)^{-d/2} (1+\gamma_{\eps,L})^d e^{-(\ell-l)^2/2\eps} \phinm. \end{align*}
	Similarly,
	\begin{align*} |\D_\eps \notine \phi (x)| &\leq (2\pi\eps)^{-d/2} (1+\gamma_{\eps,L})^d e^{-(\ell-l)^2/2\eps} \phinm \end{align*}
	Combining these results and using that $\|\cdot\|_0 \leq \|\cdot\|_{\zeta}$ we obtain what is required.
\end{proof}

This is enough for us to prove Theorem \ref{t:Delta}:
\begin{proof}[Proof of Theorem \ref{t:Delta}]
	Set
	\[  \ell = 2l = \sqrt{8\eps \min\{1,\log (2(1+\gamma_{\eps,L})^d c^{-1}(2\pi\eps)^{-d/2})\}}. \]
	From Proposition \ref{p:LInfTails} we thus have
	\[ \| (\D^M_\eps - \D_\eps) \notine \|_{H^\infty(\domain_\zeta) \to C^0(E)} \leq c. \]

	Combining this with Proposition \ref{p:LInfHeads} and restricting $\ell/\zeta \leq {{C}}_{47} \log (c^{-1}\eps^{-1})$ we have
	\begin{align*} &\Pr\left(\|(\D^M_\eps - \D_\eps) \ine \|_{H^\infty(\mathcal{E}_\zeta) \to C^0(E)} \geq 4c \phinm\right)\\
	&\qquad \leq \exp\left\{({{C}}_{48} \log \eps^{-1} + {{C}}_{49} \log c^{-1})^{d+1} \log (c^{-1}\eps^{-1})^{d} - \frac{M c^2}{4 (2\pi\eps)^{-d/2} \rhonm}\right\}. \end{align*}

	The full domain $\domain$ can be covered by $\lceil L/l\rceil^d \leq (1 + L/\sqrt{8\eps})^d$ hypercubes of side-length $l$. Thus
	 \begin{equation} \Pr\left(\|(\D^M_\eps - \D_\eps) \ine \|_{H^\infty(\mathcal{E}_\zeta) \to C^0(\domain)} \geq 4c \phinm\right) \leq \exp\left\{{{C}}_{1} (\log (2\eps c)^{-1})^{2d+1} - {{C}}_{41} M \eps^{d/2} c^2\right\}, \label{eq:StrongtoWeak}
	 \end{equation}
	 which by relabelling $4c \to c$ and $\eps \to \eps/2$, and setting ${{C}}_{2} = 2^{-d/2} {{C}}_{41}/64$ gives us \R{eq:DeltaBound}, as required.
\end{proof}

The remaining necessary ingredient for the proof of Theorem \ref{t:Operator} is a bound taking one from the weak space back into the strong space. Recall the definition of the Gaussian kernel operator \R{eq:Convolution}:
\[ \mathcal{C}_\eps\phi(x) = \int_\domain g_{\eps,L}(x-y) \phi(y)\,\dd y. \]
Then the following proposition holds:
\begin{proposition}\label{p:WeaktoStrong}
	For all $\phi \in C^0(\domain)$,
	\[ \| \mathcal{C}_\eps \phi \|_{\zeta} \leq e^{d \zeta^2/2\eps} \phinm. \]
\end{proposition}
\begin{proof}[Proof of Proposition \ref{p:WeaktoStrong}]
	Extending $\phi$ periodically to $\mathbb{R}^d$, we find that
	\begin{align*}
	\| \mathcal{C}_\eps \phi \|_{\zeta} &= \sup_{x \in \domain, z \in [-\zeta,\zeta]^d} | \C_\eps \phi(x+iz) | \\
	& \sup_{x \in \domain, z \in [-\zeta,\zeta]^d} \left|\int_{\mathbb{R}^d} (2\pi\eps)^{-d/2} e^{-(x-y+iz)^2/2\eps} \phi(y)\,\dd y\right|\\
	&\leq \sup_{x \in \domain, z \in [-\zeta,\zeta]^d} \int_{\mathbb{R}^d} (2\pi\eps)^{-d/2} e^{-\Re \sum_{j=1}^d (x_j-y_j+iz_j)^2/2\eps}\phinm\,\dd y\\
	&= \sup_{z \in [-\zeta,\zeta]^d} e^{|z|^2/2\eps} \phinm,
	\end{align*}
	giving the required result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t:Operator}]
We can decompose
\[ \D^M_\eps - \D_\eps = \mathcal{C}_{\eps/2}(\D^M_{\eps/2} - \D_{\eps/2}),\]
where we recall that $\mathcal{C}_\eps$ is convolution by a Gaussian of variance $\eps$. Combining Proposition \ref{p:WeaktoStrong} and Theorem \ref{t:Delta}, we obtain the necessary bound in the $H^\infty$ norm.
\end{proof}

\section{Convergence of the weighted operator in finite data approximation}\label{s:ParticleSinkhorn}

We now turn to the normalised operator $\P^M_\eps$. We must first bound the convergence of the function $U^M_\eps$ solving the discretised Sinkhorn problem \R{eq:SinkhornDiscretisation} converges to the continuum limit $U_\eps$ solving \R{eq:SinkhornProblem}. To apply uniform bounds on $U_\eps$ and $(I-\P_\eps)^{-1}$ in the $C^0$ norm to Hardy spaces, we will use the following proposition, whose proof is in Appendix \ref{a:ResolventProof}:
\begin{proposition}\label{p:ReciprocalZeta}
	Suppose that $\phi > 0$. Then if $\zeta = Z_0 \epsilon^{1/2}$ with
	\[Z_0 \leq \frac{\pi}{8d}(\phinm \|\phi^{-1}\|_0 \rhonm \|\rho^{-1}\|_0)^{-2},\]
	then if $\psi = 1/(\D\phi)$, the bounds in the Hardy norm \R{eq:HardyNormDefinition} hold
	\begin{align*} \| \psi \|_\zeta &\leq 2 \| \phi^{-1} \|_0\\
	  \| \psi^{-1} \|_\zeta &\leq e^{2dZ_0^2} \rhonm \| \phi\|_0. \end{align*}
\end{proposition}

As an immediate consequence we have
\begin{proposition}\label{p:SinkhornWeightZeta}
	If $\zeta = Z_0 \epsilon^{1/2}$ with $Z_0 \leq \pi (\rhonm \|\rho^{-1}\|_0 {{C}}_{22}^2)^{-2}/8d$, where ${{C}}_{22}$ is defined in Theorem \ref{t:UConvergence}, then
	\begin{equation} \|U_\eps\|_\zeta \leq 2 {{C}}_{22}^{-1}, \label{eq:WeightZetaNorm}\end{equation}
\end{proposition}

We will also find the following proposition useful:
\begin{proposition}\label{p:SinkhornResolventZeta}
	There exists a constant ${{C}}_{51}$ such that for all $\eps < \eps_0$  and $Z_0$ as in Proposition \ref{p:SinkhornWeightZeta}, then
	\begin{equation} \|(I+\P_\eps)^{-1}\|_\zeta \leq {{C}}_{51}.
	\label{eq:ResolventZetaNorm}\end{equation}
\end{proposition}

To prove this proposition we require the following result, whose proof is in Appendix \ref{a:ResolventProof}.
\begin{lemma}\label{l:Resolvent}
	There exists a constant ${{C}}_{52}$ such that for all $\eps \leq \eps_0$,
	\[ \|(I + \P_\eps)^{-1}\|_0 \leq {{C}}_{52}. \]
\end{lemma}

\begin{proof}[Proof of Proposition \ref{p:SinkhornResolventZeta}]
	We decompose
	\[ (I - \P_\eps)^{-1} = I + \P_\eps (I - \P_\eps)^{-1}. \]
	We then have for $\phi \in H^\infty(\domain_\zeta)$ that
	\[ \| \P_\eps (I - \P_\eps)^{-1} \phi \|_\zeta \leq \|U_\eps\|_\zeta \|D_\eps\|_{0 \to \zeta} \Uepsnm \|(I - \P_\eps)^{-1}\|_0 \|\phi\|_0, \]
	which by an application of Proposition \ref{p:SinkhornWeightZeta} and Lemma \ref{l:Resolvent} gives
	\[ \| \P_\eps (I - \P_\eps)^{-1} \phi \|_\zeta \leq 2{{C}}_{22}^2 e^{2dZ_0^2} {{C}}_{52} \|\phi\|_0. \]
	Using that $\|\cdot\|_0 \leq \|\cdot\|_\zeta$ we obtain the required result.
\end{proof}

We can now prove convergence of the Sinkhorn weight as the number of particles $M \to \infty$:
\begin{lemma}\label{l:Sinkhorn}
	Suppose $Z_0$ is as in Proposition \ref{p:SinkhornWeightZeta}. There exist constants ${{C}}_{9}, {{C}}_{53}$
	such that if $\delta \leq {{C}}_{9}$ then
	\[ \| U^M_\eps - U_\eps\|_\zeta,\, \| Y^M_\eps - Y_\eps\|_0,\, \| (Y^M_\eps)^{-1} - (Y_\eps)^{-1}\|_0 \leq {{C}}_{53} \delta, \]
	where $\zeta = Z_0 \eps^{1/2}$.
\end{lemma}

\begin{proof}[Proof of Lemma \ref{l:Sinkhorn}]
	We can rewrite \R{eq:SinkhornProblem} and \R{eq:SinkhornDiscretisation} as
	\begin{align*}
	U_\eps(x)\, (\D_{\eps}U_\eps)(x) &\equiv 1\\
	U^M_\eps(x) \, (\D^M_{\eps}U^M_\eps)(x) &\equiv 1.
	\end{align*}
	If for $\theta \in [0,1]$ we set
	\[ \D^\theta_\eps := (1-\theta)\D_{\eps} + \theta\D^M_{\eps} \]
	then we obtain a one-parameter family of Sinkhorn weight functions $U^\theta_\eps$ solving
	\begin{equation} U^\theta_\eps(x)\, (\D^\theta_{\eps}U^\theta_\eps)(x) \equiv 1. \label{eq:SinkhornContinuum}\end{equation}
	The existence and uniqueness of the $U^\theta_{\eps}$ follow from the positivity of the operator $\D^\theta_{\eps}$, on $L^\infty(\domain)$ for $\theta\in[0,1)$ and on $L^\infty(\{x^i\}_{i=1\ldots M})$ for $\theta = 1$.

	Furthermore, because
	\[\frac{\dd}{\dd\theta} \D^\theta_\eps = \D^M_\eps - \D_{\eps} \]
	is a bounded operator on $H^\infty_\zeta$, we can apply the implicit function theorem to \R{eq:SinkhornContinuum} as long as $U^\theta_\eps$ stays in $H^\infty_\zeta$, so that
	\[ \frac{\dd}{\dd\theta} \log U^\theta_\eps = -(I + U^\theta_\eps \D^\theta_{\eps}U^\theta_\eps)^{-1} U^\theta_\eps(x) \left((\D^M_\eps - \D_{\eps})U^\theta_\eps\right)(x). \]

	We have from Propositions \ref{p:SinkhornWeightZeta} and \ref{p:SinkhornResolventZeta} that $\|U_\eps\|_\zeta \leq 2{{C}}_{22}$, and $\|(I-\P_\eps)^{-1}\|_\zeta \leq {{C}}_{51}$, and from Theorem \ref{t:Operator} that $\|\D^M_\eps - \D_\eps\|_\zeta \leq e^{2d Z_0^2} \delta$. Note that since $\P_\eps$ has $1$ as an eigenvalue, ${{C}}_{51} \geq 1/2$.

	If $ B(\theta) := \| \log U^\theta_{\eps} - \log U_\eps \|_\zeta$, then
	\[ B'(\theta) \leq \left\| \frac{\dd}{\dd\theta} \log U^\theta_\eps\right\|_\zeta
	\leq {{C}}_{51} (1 - {{C}}_{51} \|U^\theta_\eps \D^\theta_{\eps}U^\theta_\eps - \P_\eps \|_\zeta)^{-1} e^{2d Z_0^2} \delta \|U^\theta_\eps\|_\zeta.
	\]
	Because $\|U^\theta_\eps\|_\zeta \leq 2{{C}}_{22} e^{B(\theta)}$,
	\[ B'(\theta)
	\leq \frac{4{{C}}_{22}^2 {{C}}_{51} e^{2d Z_0^2} \delta e^{2B(\theta)}}{1 - 4{{C}}_{22}^2 {{C}}_{51} (\|\D^\theta_{\eps}\|_\zeta B(\theta)(e^{2B(\theta)}+e^{B(\theta)} + e^{2d Z_0^2} \delta e^{B(\theta)}) )}
	\]
	and because $\|\D^\theta_{\eps}\|_\zeta \leq \|\D_{\eps}\|_\zeta + \theta e^{2d Z_0^2} \delta \leq 1 + e^{2d Z_0^2} \delta$,
	\[ B'(\theta)
	\leq \frac{4{{C}}_{22}^2 {{C}}_{51} e^{2d Z_0^2} \delta e^{2B(\theta)}}{1 - 4{{C}}_{22}^2 {{C}}_{51} (2(1+e^{2d Z_0^2} \delta)B(\theta)e^{2B(\theta)} + e^{2d Z_0^2} \delta e^{B(\theta)}) )} .
	\]

	Thus, as long as $B(\theta) \leq \min\{e^{2d Z_0^2} \delta, {{C}}_{54}\}$ and $\delta \leq {{C}}_{55}/(4{{C}}_{22}^2 {{C}}_{51} e^{2d Z_0^2}) =: {{C}}_{59}$ for some fixed constants ${{C}}_{54},{{C}}_{55},{{C}}_{56}$,
	\[ B'(\theta) \leq {{C}}_{56} 4{{C}}_{22}^2 {{C}}_{51} e^{2d Z_0^2} \delta \]
	and thus
	\[  \|\log U_\eps - \log U^M_\eps\|_\zeta \leq B(1) \leq {{C}}_{56} 4{{C}}_{22}^2 {{C}}_{51} e^{2d Z_0^2} \delta. \]

	Furthermore, for some fixed constant ${{C}}_{57}$,
	\begin{align*}\|U^M_\eps - U_\eps \|_\zeta &\leq \|U_\eps\|_\zeta \left(e^{\|\log U_\eps - \log U^M_\eps\|_\zeta} - 1\right)\\
	& \leq {{C}}_{57} 8{{C}}_{22}^3 {{C}}_{51} e^{2d Z_0^2} \delta\\
	& \leq {{C}}_{57} 8 {{C}}_{22}^3 {{C}}_{51} e^{2 d Z_0^2} \delta =: {{C}}_{58} \delta,
	\end{align*}
	as required.

	To prove the second part, we use the definition of $Y^{(M)}_\eps$ in \R{eq:Y} to say that
	\[ Y^M_\eps - Y_\eps = (\D^M_{\eps/2} - \D_{\eps/2}) U^M_\eps - \D_{\eps/2} (U^M_\eps - U_\eps) \]
	and so
	\[ \| Y^M_\eps - Y_\eps \|_0 \leq (2{{C}}_{22} + {{C}}_{59} + {{C}}_{58})\delta =: {{C}}_{60} \delta. \]
	Furthermore,
	\[ (Y^M_\eps)^{-1} - (Y_\eps)^{-1} = \frac{Y_\eps^{-2} (Y^M_\eps - Y_\eps)}{1-Y_\eps^{-1} (Y_\eps - Y^M_\eps)} \]
	and so using that
	\[ \|Y_\eps^{-1}\|_0 \leq \|\rho^{-1}\|_0 \|U_\eps^{-1}\|_0 \leq \|\rho^{-1}\|_0 \|\rho U_\eps\|_0 = \|\rho^{-1}\|_0 \rhonm {{C}}_{22}, \]
	we have that provided that $ \delta < \min\{{{C}}_{59},({{C}}_{60} \half \|\rho^{-1}\|_0 \rhonm {{C}}_{22})^{-1}\} =: {{C}}_{9}$,
	\[ \|(Y^M_\eps)^{-1} - (Y_\eps)^{-1}\|_0 \leq 2(\|\rho^{-1}\|_0 \rhonm {{C}}_{22})^2 {{C}}_{60}\delta =: {{C}}_{53}\delta. \]
	Readjusting ${{C}}_{53} = \max\{{{C}}_{58},{{C}}_{60},{{C}}_{53}\}$, we have what is required.
\end{proof}

The convergence of the Sinkhorn-weighted operator then follows in Theorem \ref{t:WeightedOperatorConvergence}, which we prove here:
\begin{proof}[Proof of Theorem \ref{t:WeightedOperatorConvergence}]
	We can decompose
	\[ \P^M_\eps - \P_\eps = U^M_\eps (\D^M_\eps - \D_{\eps})U^M_\eps + U^M_\eps\D_\eps (U^M_\eps - U_\eps) + (U^M_\eps - U_\eps)\D_\eps U_\eps.\]
	Using Lemma \ref{l:Sinkhorn}, Propositions \ref{p:SinkhornWeightZeta} and \ref{p:SinkhornResolventZeta} and that $\|\D_{\eps}\|_\zeta \leq \rhonm$ we have that
	\begin{align*} \| \P^M_\eps - \P_\eps \|_\zeta &\leq \big((2{{C}}_{22} + {{C}}_{53} {{C}}_{9})^2 {{C}}_{53}  + (2{{C}}_{22} + {{C}}_{53} {{C}}_{9}) \rhonm {{C}}_{53}  + \\
	&\qquad {{C}}_{53}  \rhonm 2{{C}}_{22} \big) e^{2 dZ_0^2} \delta, \\
	& =: {{C}}_{10}\delta \end{align*}
	for some constant ${{C}}_{10}$.

	The corresponding bounds for the half-step operators $\J^M_\eps,\, \K^M_\eps$ and the semi-conjugate operator $\M^M_{\eps,1}$ arise similarly, with an appropriate adjustment of ${{C}}_{10}$; this extends to general $\M^M_{\eps,n} = (\M^M_{\eps,1})^n$ by using that $\M^M_{\eps,1}, \M_{\eps,n}$ are row-stochastic and thus have unit $C^0$ norm.
\end{proof}

config: arxiv