# Analysis of the feedback particle filter with diffusion map based
approximation of the gain

Sahani Pathiraja  Institute of Mathematics, University of Potsdam, Germany
Wilhelm Stannat  Institute of Mathematics, TU Berlin, Germany  Bernstein
Center for Computational Neuroscience, Berlin, Germany

###### Abstract

Control-type particle filters have been receiving increasing attention over
the last decade as a means of obtaining sample based approximations to the
sequential Bayesian filtering problem in the nonlinear setting. Here we
analyse one such type, namely the feedback particle filter and a recently
proposed approximation of the associated gain function based on diffusion
maps. The key purpose is to provide analytic insights on the form of the
approximate gain, which are of interest in their own right. These are then
used to establish a roadmap to obtaining well-posedness and convergence of the
finite $N$ system to its mean field limit. A number of possible future
research directions are also discussed.

Keywords. nonlinear filtering, diffusion maps, Poisson equation, propagation
of chaos, stochastic analysis
AMS(MOS) subject classifications. 60H10, 35Q93, 35J05, 60J60, 60J27, 62M05

##  1 Introduction

The filtering problem is concerned with the following system

$\displaystyle dS_{t}$ $\displaystyle=\mathcal{M}(S_{t})dt+dV_{t},$
$\displaystyle dZ_{t}$ $\displaystyle=h(S_{t})dt+dW_{t},$

where $S_{t}\in\mathbb{R}^{d}$ and $Z_{t}\in\mathbb{R}$ with
$\mathcal{M}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and
$h\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$.
$V_{t}$ and $W_{t}$ are independent Wiener processes and we define the
filtration $\mathcal{Z}_{t}\mathrel{\mathop{\mathchar
58\relax}}=\sigma(Z_{s}\mathrel{\mathop{\mathchar 58\relax}}s\leq t)$. The
goal of the filtering problem is to estimate the conditional distribution
$P\\{S_{t}\in A|\mathcal{Z}_{t}\\}$ (also referred to as the filtering
distribution or posterior distribution in Bayesian terminology). The Kushner-
Stratonovich SPDE describes the time evolution of the filtering distribution.
It is well-known that there is no closed-form exact description of the
conditional distribution for general $\mathcal{M},h$, with the exception of
the linear case with Gaussian initial condition, for which the Kalman-Bucy
filter provides the exact posterior first and second central moments. The
Monte-Carlo based ensemble Kalman filter (EnKF) [Eve06], [Eve03] [EvLP00] has
been shown to provide computationally robust posterior samples for high
dimensional nonlinear applications, which has thus spurred the theoretical
analysis of both the EnKf (i.e. in discrete time) [LMT11] [TMK16] [MT16] and
ensemble Kalman-Bucy filter (EnKBF) (i.e. in continuous time time) [BR12],
[dWRS18], [dWT20], [LS21] [SS16] [BdP18] [dKT17]. These references are by no
means exhaustive and the literature on this topic continues to expand. Despite
its desirable computational properties particularly in the small ensemble
setting, the EnKF and EnKBF do not produce samples from the true posterior,
even as the number of particles $N\rightarrow\infty$ due to the linear update
approximation. Traditional sequential Monte Carlo approaches (also known as
particle filters) are asymptotically consistent but have had limited practical
application due to know well-known weight degeneracy issues even in problems
of moderate dimension (e.g. $d<100$). Over the last decade, there has been
increasing attention given to asymptotically consistent controlled filtering
methods in the like of [CX10], [Rei10] and in particular, the feedback
particle filter (FPF) [YMM13], [LMMR15], [YMM11]. These methods involve
constructing a McKean-Vlasov process to guide the evolution of the particles
in state space, thereby combining the robustness advantages of the EnKBF and
the consistency properties of traditional SMC. The process is constructed to
ensure that the conditional distribution of the process at time $t$ given
$\mathcal{Z}_{t}$ coincides exactly with the solution of the Kushner-
Stratonovich equation. The general form of this process is [PRS21]

$\displaystyle d\tilde{X}_{t}$
$\displaystyle=\mathcal{M}(\tilde{X}_{t})dt+d\tilde{V}_{t}+K(\tilde{X}_{t},\tilde{\rho}_{t})\left(dZ_{t}-\frac{1}{2}(h(\tilde{X}_{t})+\hat{\tilde{h}}_{t})dt\right)$
(1.1) $\displaystyle+\frac{1}{2}(\nabla
K(\tilde{X}_{t},\tilde{\rho}_{t}))^{T}K(\tilde{X}_{t},\tilde{\rho}_{t})dt+\mathcal{J}(\tilde{X}_{t},\tilde{\rho}_{t})dt,$

where $\mathcal{J}$ and $K$ satisfy

$\displaystyle\nabla\cdot(\tilde{\rho}_{t}\mathcal{J}(x,\tilde{\rho}_{t}))$
$\displaystyle=0,$ (1.2)
$\displaystyle\nabla\cdot(\tilde{\rho}_{t}K(x,\tilde{\rho}_{t}))$
$\displaystyle=-(h-\hat{\tilde{h}}_{t})\tilde{\rho}_{t},$ (1.3)

respectively, $\tilde{\rho}_{t}$ indicates the conditional density of the
process given $\mathcal{Z}_{t}$ and $\hat{\tilde{h}}_{t}=\int
h(x)\tilde{\rho}_{t}(x)dx$. The non-uniqueness of solutions of (1.3) gives
rise to many formulations established in the literature, and can be further
analysed to obtain new formulations with desirable analytic and numerical
properties. A further unifying analysis of these filters can be found in
[PRS21]. The focus of this paper is the FPF, a special case of (1.1) which can
be obtained for the choice that $K$ is of gradient type. More specifically,
the exact FPF equations are

$\displaystyle d\tilde{X}_{t}^{F}$
$\displaystyle=\mathcal{M}(\tilde{X}_{t}^{F})dt+d\tilde{V}_{t}^{F}+K(\tilde{X}_{t}^{F},{\rho}_{t})\left(dZ_{t}-\frac{1}{2}(h(\tilde{X}_{t}^{F})+\hat{{h}}_{t})dt\right)$
(1.4) $\displaystyle+\frac{1}{2}(\nabla
K(\tilde{X}_{t}^{F},{\rho}_{t}))^{T}K(\tilde{X}_{t}^{F},{\rho}_{t})dt,$

where $K=\nabla\phi$ and $\phi$ satisfies the weighted Poisson equation

$\displaystyle-\Delta_{\rho_{t}}\phi(x)\mathrel{\mathop{\mathchar
58\relax}}=-\frac{1}{\rho_{t}}\nabla\cdot(\rho_{t}\nabla\phi(x)=(h-\hat{h}_{t}),$
(1.5)

for strictly positive densities $\rho_{t}$, where $\rho_{t}$ refers to the
conditional density of the process $\tilde{X}_{t}^{F}$. $K$ is typically
referred to as the gain function, due to its similarity with the Kalman gain
in the Kalman filter. The well-posedness of (1.5) under general conditions on
$\mathcal{M}$ and $h$ in unbounded domains is a largely open research
question, with some first steps in [PRS21] and [LMMR15]. A computational
bottleneck of this approach is in obtaining an approximate solution to (1.5).
An attractive feature of the FPF is that adopting the so-called constant gain
approximation in (1.4) leads to the enKBf. Various methods have been proposed
for higher order approximations; see for instance using Galerkin based PDE
solution methods with POD adjustment for high dimensional applications see
e.g. [BG16] and also using dynammic programming [RDM16] and also more recently
based on the active area of using neural networks for approximating solutions
of high dimensional PDEs [OTM20]. Here we focus on the recently proposed
method of [TMM20] which makes use of diffusion maps. Diffusion maps [CL06] is
a manifold learning technique which uses the eigenvectors of the diffusion
operator on a data set to determine a typically lower dimensional underlying
manifold from which the data has been sampled. This is done by first
constructing a family of anisotropic diffusion processes which are the limits
of a graph Laplacian jump process. This family is parameterised by $\alpha$
which reflects the influence of the density of the data points in determining
the infinitessimal transitions of the diffusion. The key point here is to
extend the classical ideas of using the normalised graph Laplacian to cluster
data for a graph to the Euclidean space setting by making use of the
arrangement or density of the data points. Alternative normalisations are
possible and advantageous, see e.g. an approach based on Sinkhorn weights in
[WR21]. For further details on diffusion maps, see in particular the seminal
work of [CL06].

###  1.1 Diffusion map based approximation of the gain

The aforementioned properties of diffusion maps make it an ideal candidate for
developing a sample based approximation of $\phi$. [TMM20] uses the diffusion
map approach in a novel way by starting from the semigroup formulation of the
Poisson equation (1.5). Their algorithm is summarised in the remainder of this
section, we refer to [TMM20] for further details and derivations. The
semigroup formulation of (1.5) is given by

$\displaystyle\phi=P_{\epsilon}\phi+\int_{0}^{\epsilon}P_{s}(h-\hat{h})ds,$
(1.6)

where $\\{P_{\epsilon}\\}_{\epsilon\geq 0}$ is the Markov semigroup associated
to the weighted Laplacian $\Delta_{\rho_{t}}$, which must be approximated. In
other words, $P_{s}$ is the transition operator of a Langevin diffusion with
invariant density given by $\rho_{t}$. Adopting $\alpha=\frac{1}{2}$ in the
diffusion map method corresponds to a Markov chain approximating the Langevin
diffusion of interest [CL06], with anisotropic transition kernel denoted by
$k_{\epsilon}$ and

$\displaystyle k_{\epsilon}(x,y)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\tilde{g}_{\epsilon}(x,y)}{(\int\tilde{g}_{\epsilon}(x,z)\rho_{t}(z)dz)^{1/2}(\int\tilde{g}_{\epsilon}(y,z)\rho_{t}(z)dz)^{1/2}},$

where

$\displaystyle\tilde{g}_{\epsilon}(x,y)\mathrel{\mathop{\mathchar
58\relax}}=(4\pi\epsilon)^{-d/2}\exp(-\frac{1}{4\epsilon}|x-y|^{2}),$ (1.7)

is a Gaussian kernel with bandwidth matrix $2\epsilon I_{d}$ where $I_{d}$
denotes the identity matrix of size $d$. This can be used to obtain the
following approximation of the transition operator $P_{\epsilon}$, denoted by
${T}_{\epsilon}$,

$\displaystyle{T}_{\epsilon}f(x)\mathrel{\mathop{\mathchar 58\relax}}=\int
p_{\epsilon}(x,y)f(y)dy,$ (1.8)

where

$\displaystyle p_{\epsilon}(x,y)\mathrel{\mathop{\mathchar
58\relax}}=\frac{k_{\epsilon}(x,y)\rho_{t}(y)}{k_{\epsilon}*\rho_{t}(x)},$

where we use the shorthand notation
$k_{\epsilon}*\rho_{t}(x)\mathrel{\mathop{\mathchar 58\relax}}=\int
k_{\epsilon}(x,y)\rho_{t}(y)dy$, also in the remainder of the article.
Additionally, we will use the following equivalent simplified form throughout
the article

$\displaystyle
p_{\epsilon}(x,y)=\frac{g_{\epsilon}(x,y)\nu_{\epsilon}(y)}{g_{\epsilon}*\nu_{\epsilon}(x)},$
(1.9)

where $g_{\epsilon}(x,y)\mathrel{\mathop{\mathchar
58\relax}}=\exp(-\frac{1}{4\epsilon}\mathinner{\\!\left\lVert
x-y\right\rVert}^{2})$ is the unnormalised Gaussian kernel and

$\displaystyle\nu_{\epsilon}(y)=\frac{\rho_{t}(y)}{(g_{\epsilon}*\rho_{t}(y))^{1/2}}.$
(1.10)

An infinite dimensional, diffusion map based approximation of $\phi$, denoted
by $\phi_{\epsilon}$, is then obtained as a solution of

$\displaystyle\phi_{\epsilon}={T}_{\epsilon}\phi_{\epsilon}+\epsilon(h-\hat{h}_{\epsilon});\quad\hat{h}_{\epsilon}\mathrel{\mathop{\mathchar
58\relax}}=\int h(x)\pi_{\epsilon}(x)dx,$ (1.11)

where $\pi_{\epsilon}(x)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\rho_{t}(x)k_{\epsilon}*\rho_{t}(x)}{\int\rho_{t}(x)k_{\epsilon}*\rho_{t}(x)dx}$
is the invariant density associated with the Markov operator ${T}_{\epsilon}$.
Now given $N$ $\rho_{t}$-distributed $\mathbb{R}^{d\times 1}$-valued random
samples denoted by $\\{X^{i}\\}_{i\in\\{1,2,\cdots,N\\}}$, a finite
dimensional approximation of ${T}_{\epsilon}$ is constructed as an $N\times N$
Markov matrix denoted by $T$ with $(i,j)$th entry given by

$\displaystyle T_{ij}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{k_{\epsilon}^{N}(X^{i},X^{j})}{\sum_{l}{k}_{\epsilon}^{N}(X^{i},X^{l})},$
(1.12a)
$\displaystyle=\frac{\tilde{q}_{\epsilon}(X^{i},X^{j})}{{\sum_{l}\tilde{q}_{\epsilon}(X^{i},X^{l})}},$
(1.12b)

where

$\displaystyle\quad{k}_{\epsilon}^{N}(X^{i},X^{j})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{\tilde{g}_{\epsilon}(X^{i},X^{j})}{(\sum_{l}\tilde{g}_{\epsilon}(X^{j},X^{l}))^{1/2}(\sum_{l}\tilde{g}_{\epsilon}(X^{i},X^{l}))^{1/2}},$
$\displaystyle\tilde{q}_{\epsilon}(X^{i},X^{j})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{g_{\epsilon}(X^{i},X^{j})}{(\sum_{l}g_{\epsilon}(X^{j},X^{l}))^{1/2}}.$

In the remainder of the article, we will adopt the shorthand notation

$\displaystyle s_{\epsilon}^{i}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{l}\tilde{q}_{\epsilon}(X^{i},X^{l}).$ (1.13)

Furthermore, $T$ is a reversible Markov matrix with strictly positive entries
and a unique stationary distribution (see proposition 4.1 [TMM20]). $T$ is
then used to construct a finite dimensional approximation of the fixed point
equation (1.11),

$\displaystyle\Phi=T\Phi+\epsilon({\bf h}-\hat{h}^{N}\mathbf{1}),$ (1.14)

where $\Phi\in\mathbb{R}^{N\times 1}$, ${\bf h}\mathrel{\mathop{\mathchar
58\relax}}=[h(X^{1}),h(X^{2}),\cdots,h(X^{N})]^{T}$,
$\mathbf{1}\in\mathbb{R}^{N\times 1}$ is a vector of 1’s and
$\hat{h}^{N}=\sum_{i=1}^{N}\pi_{i}h(X^{i})$ where
$\pi_{i}=\frac{\sum_{l}\tilde{q}_{\epsilon}(X^{i},X^{l})}{\sum_{j}\sum_{l}\tilde{q}_{\epsilon}(X^{j},X^{l})}$
is the stationary distribution of the Markov matrix $T$. The solution of
(1.14) is denoted by $\Phi^{\infty}$ and it holds that

$\displaystyle\Phi^{\infty}_{j}-\Phi^{\infty}_{i}=\epsilon(h(X^{j})-h(X^{i}))+\epsilon\sum_{n=1}^{\infty}[T^{n}{\bf
h}]_{j}-[T^{n}{\bf h}]_{i}.$ (1.15)

Taking the derivative of an intermediate approximation of (1.11) given by

$\displaystyle\phi_{\epsilon}^{(N)}(x)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sum_{i}\tilde{q}_{\epsilon}(x,X^{i})}\sum_{j=1}^{N}\tilde{q}_{\epsilon}(x,X^{j})\Phi_{j}+\epsilon(h(x)-\hat{h}^{N}),$
(1.16)

(see Appendix C in [TMM20]) leads to the following finite dimensional
approximation of the gain $K=\nabla\phi$

$\displaystyle
K_{\epsilon}(X^{i},\rho^{N})=\frac{1}{2\epsilon}\sum_{j}\sum_{k}T_{ij}T_{ik}(r_{j}-r_{k})X^{j},$

where

$\displaystyle r_{j}\mathrel{\mathop{\mathchar
58\relax}}=\Phi_{j}^{\infty}+\epsilon h(X^{j}),$ (1.17)

and $\rho^{N}$ denotes the empirical measure of the particles; which can be
equivalently expressed as

$\displaystyle K_{\epsilon}(X^{i},\rho^{N})$
$\displaystyle=\sum_{j=1}^{N}s_{ij}X^{j},$ (1.18a)
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2\epsilon}\sum_{j}T_{ij}X^{j}r_{j}-\frac{1}{2\epsilon}\left(\sum_{j}T_{ij}X^{j}\right)\left(\sum_{j}T_{ij}r_{j}\right),$
(1.18b)

which has the benefit of making explicit the weighted cross-covariance type
structure, thereby showing the resemblance to the Kalman gain in the enKBf. An
attractive feature of the diffusion map based approximation of the gain is
that it leads to a bridging between the constant gain approximation (i.e. the
enKBf) and the exact filter via the $\epsilon$ parameter. More specifically,
as $\epsilon\rightarrow\infty$, the approximation (1.18b) approaches the
ensemble Kalman gain matrix. For a precise explanation, see [TMM20].

###  1.2 Problem statement

Throughout the article, we consider the following $N$ particle process
approximating (1.4) excluding the Stratonovich term, with the diffusion map
based approximation of $K$ from [TMM20],

$\displaystyle
dX_{t}^{i}=\mathcal{M}(X_{t}^{i})dt+dV_{t}^{i}+K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\left(dZ_{t}-\frac{1}{2}(h(X_{t}^{i})+\hat{h}_{t}^{N})dt\right),$
(1.19)

where $\rho_{t}^{N}$ denotes the empirical measure of the $N$ particle process
at time $t$ and $K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})$ is a finite dimensional
diffusion map based approximation of $K$ defined in (1.18b). The Stratonovich
term is neglected in (1.19) primarily to simplify the analysis; the influence
of this term in practical applications will generally be quite small for large
$N$ and with Lipschitz $\mathcal{M}$ and $h$ where the distributions are close
to the Gaussian setting (since the gain $K$ will be close to constant in $x$
so that the derivative term is close to zero.)

The main goal of this article is two-fold; 1) to determine the well-posedness
of the finite $N$ particle system (1.19); and 2) to establish the existence of
a mean-field limit of (1.19) as $N\rightarrow\infty$ for fixed $\epsilon>0$.
In order to address the second point, we first propose that the limiting form
of (1.19) can be described by the following McKean-Vlasov process

$\displaystyle
d\bar{X}_{t}=\mathcal{M}(\bar{X}_{t})dt+d\bar{V}_{t}+\bar{K_{\epsilon}}(\bar{X}_{t},\bar{\rho}_{t})\left(dZ_{t}-\frac{1}{2}(h(\bar{X}_{t})+\hat{\bar{h}}_{t})dt\right),$
(1.20)

where $\bar{\rho}_{t}$ indicates the density function associated to the
conditional law of the process given the observations up to time $t$ and
$\hat{\bar{h}}_{t}=\int h(x)\rho_{t}(x)dx$. The gain function
$\bar{K}_{\epsilon}$ is the mean-field form of (1.18b) and is defined as

$\displaystyle\bar{K}_{\epsilon}(x,\bar{\rho}_{t})\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2\epsilon}\int\mathcal{R}_{\epsilon}(y)yp_{\epsilon}(x,y)dy-\frac{1}{2\epsilon}\left(\int
yp_{\epsilon}(x,y)dy\right)\left(\int\mathcal{R}_{\epsilon}(y)p_{\epsilon}(x,y)dy\right),$

where $\mathcal{R}_{\epsilon}(y)\mathrel{\mathop{\mathchar
58\relax}}=\phi_{\epsilon}(y)+\epsilon h(y)$ and $\phi_{\epsilon}(y)$ is the
solution of the fixed point equation (1.11) and the transition density
$p_{\epsilon}(x,y)$ is as defined in (1.9), but with $\rho_{t}$ replaced by
$\bar{\rho}_{t}.$

The well-posedness of the diffusion map approximation of $\phi$ as well as the
convergence in both $\epsilon\rightarrow 0$ (bias) and $N\rightarrow 0$
(variance) was investigated in [TMM20] for the case of iid samples from a
density of the form $\rho(x)=\exp(-V(x))$ with
$V(x)=\frac{1}{2}(x-m)^{T}\Sigma(x-m)+w(x)$ where $w(x)$ is smooth and
uniformly bounded. The convergence of $\phi_{\epsilon}^{(N)}$ as defined in
(1.16) to $\phi_{\epsilon}$ as $N\rightarrow\infty$ was established for
densities supported on compact domains.

###  1.3 Statement of the main results

######  Theorem 1.1.

Well-posedness of the finite $N$ system. Assume
$\mathcal{M}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is globally Lipschitz with
Lipschitz constant $L_{M}$ and $h\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$ is uniformly bounded and
globally Lipschitz with Lipschitz constant $L_{h}$. Then the $N$-approximate
interacting particle system (1.19) with finitely many iterates used to solve
(1.14) posesses a unique strong solution for all $t>0$.

The proof of Theorem 1.1 is given in Section 3. As will be made clear there,
the well-posedness result does not hold uniformly in $N$ and well-posedness of
the mean field process (1.20) is non-trivial due to the fact that the
coefficients have non-Lipschitz dependence on the measure argument. However,
the properties of $\bar{K}_{\epsilon}$ with respect to the spatial argument
established in Section 2 indicate a promising pathway to well-posedness for
processes defined on unbounded domains, assuming that the conditional density
of the process has sufficiently fast tail decay. We capture this more
precisely via log-concavity, see Assumption 2.1. The next main result relates
to the convergence of the finite $N$ process to its mean field limit.
Propagation of chaos type results have been established for the enKBf in the
linear case [dKT17] and also more general ensemble square root filters [LS20].
The covariance structure of the gain lends itself well to such analysis. The
weighted covariance structure of (1.18b)leads to several complications; our
main result is therefore a trajectorial propagation of chaos up to a stopping
time. That is,

######  Theorem 1.2.

Trajectorial propagation of chaos. Given $h$ and $\mathcal{M}$ globally
Lipschitz with Lipschitz constants $L_{h}$ and $L_{M}$ respectively, $h$
uniformly bounded, and for (1.14) solved exactly and $\epsilon>0$, the
following mean square error convergence result holds up to a stopping time
$\zeta_{\delta}^{N}>0$ for large enough $N$,

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i=1}^{N}\left|X_{t}^{i}-\bar{X}_{t}^{i}\right|^{2}\right]\leq{\frac{C}{N}},$
(1.21)

where

$\displaystyle C=f(\epsilon,d,L_{M},L_{h},\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}),$

and $X_{t}^{i}$ is the $i$th particle solution of (1.19) with iid initial
conditions $\\{X_{0}^{i,N}\\}_{i\in\\{1,2,\cdots N\\}}$ and
$\\{\bar{X}_{t}^{i}\\}_{i\in\\{1,2,\cdots N\\}}$ are $N$ independent solutions
of (1.20) with the same Brownian motions $\\{V_{t}^{i}\\}$ and initial
conditions used to generate $\\{{X}_{t}^{i}\\}_{i\in\\{1,2,\cdots N\\}}$.

The precise form of the stopping time and conditions for large enough $N$ are
given in Section 4, along with the proof of the above theorem in Section 4.2.
The stopping time arguments used in the proof of this theorem allow us to
consider the exact solution to (1.14), unlike the $L$-approximate solution
considered in Theorem 1.1. The precise form of the constant $C$ can also be
found in Section 4.2, where it can be seen that it scales rather poorly with
the dimension $d$, although this maybe used to specify a criterion for
choosing $\epsilon$. The complicated structure of the diffusion map
approximation to the gain function means that standard propagation of chaos
techniques for interacting particle systems e.g. [BGM10] are not directly
applicable here, leading to a non-trivial analysis. The analysis also differs
from that used to obtain standard stability results in discrete time particle
filters. There are several open questions which should be explored in future
research, in addition to the well-posedness mentioned above. In particular,
extending Theorem 1.2 to arbitrary time intervals $[0,T]$ for any $T>0$ by way
of apriori estimates on $\bar{X}$ and $X$ to ensure non-explosion in finite
time (i.e. a finite time well-posedness result). As discussed above, the well-
posedness of the mean field process remains yet to be established and is an
open problem in the literature [LS20]. Such localisation arguments usually
allow for convergence in probability, although $L^{2}$ convergence as in
Theorem 1.1 may be possible using similar arguments via Moore Osgood theorem
as used in [Pat20]. Finally, it would be of course pertinent to consider the
gain function approximation in the exact filter (1.4) and to consider the
simultaneous limits $\epsilon\rightarrow 0$ and $N\rightarrow\infty$. The
precise relation between $\epsilon$ and $N$ would most likely need to be
carefully controlled, as indicated by the bias variance discussion in [TMM20].
We note that the convergence of the fixed point equation on unbounded domains
is still an open question in general[TMM20] and would also need to be
established. On a more practical note, it may also be worth considering
variable bandwidth kernels for which several advantages have been established,
see e.g. [BH16].

The remainder of the article is structured as follows. Section 2 provides some
analytic properties of the mean field gain $K_{\epsilon}(x,\mu)$ subject to
certain conditions on $\mu$. Section 3 examines well-posedness of the $N$
interacting particle equations for fixed $N$, i.e. the proof of the first main
result. Finally, Section 4 contains the proof of second main result along with
several important lemmas necessary for proving Theorem 1.2.

##  2 Analytic properties of $\bar{K}_{\epsilon}(x,\mu)$

This section presents some insights on properties of the mean field form of
the gain, $\bar{K}_{\epsilon}$, in unbounded domains by making use of a log-
concavity assumption. They are mainly of independent interest and maybe
utilised for obtaining propagation of chaos type results as in Section 4
without the use of stopping times. The exponential tail decay of the
conditional density ensured by the aforementioned assumption plays an
important role in uniformly bounding the gain in the spatial argument, which
is a rather remarkable result given the covariance type structure of the gain
and without boundedness assumptions on $h$. More specifically, in this section
we assume the following. Note that we use the notation $\succcurlyeq$ to
indicate a matrix inequality, i.e. $A\succcurlyeq B$ means $(A-B)$ is positive
semidefinite.

######  Assumption 2.1.

The conditional density of the mean field process (1.20) at time $t$ is
strongly log-concave, i.e. it takes the form $\exp(-\mathcal{V}_{t}(x))$ with

$\displaystyle\nabla_{x}^{2}\mathcal{V}_{t}\succcurlyeq c_{v}I,$

for some $c_{v}>0$.

######  Lemma 2.1.

Log-concavity of the transition kernel Under Assumption 2.1, it holds that the
transition kernel $p_{\epsilon}(x,y)$at time $t$ is strongly log-concave in
$y$ for all $x\in\mathbb{R}^{d}$. That is,
$p_{\epsilon}(x,y)\propto\exp(-\mathcal{G}_{t}(x,y))$ with

$\displaystyle\nabla_{y}^{2}\mathcal{G}_{t}\succcurlyeq c_{g}I,$

where

$\displaystyle c_{g}=\frac{1}{4\epsilon}+c_{v}.$ (2.1)

###### Proof.

The unnormalised form of $p_{\epsilon}(x,y)$ can be rewritten as

$\displaystyle g_{\epsilon}(x,y)\nu_{\epsilon}(y)$
$\displaystyle=\exp\left(-\frac{1}{4\epsilon}|x-y|^{2}-\mathcal{V}_{t}(y)-\frac{1}{2}\log(g_{\epsilon}*\bar{\rho}_{t}(y))\right)$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\exp(-\mathcal{G}_{t}(x,y)),$

and

$\displaystyle\nabla_{y}^{2}\mathcal{G}_{t}=\frac{1}{2\epsilon}I+\nabla_{y}^{2}\mathcal{V}_{t}(y)+\frac{1}{2}\left(\frac{1}{4\epsilon^{2}}Cov_{\omega(y,z)dz}(z)-\frac{1}{2\epsilon}I\right),$

where $\omega(y,z)\propto\exp(-F_{t}(y,z))$ and
$F_{t}(y,z)=\frac{1}{4\epsilon}|y-z|^{2}+\mathcal{V}_{t}(z)$. Differentiation
under the integral sign is justified by the same arguments as in the proof of
Theorem 4.2 in [BL76]. Due to the positive-definiteness of
$Cov_{\omega(y,z)dz}(z)$, it holds that

$\displaystyle\nabla_{y}^{2}\mathcal{G}_{t}$
$\displaystyle\succcurlyeq\frac{1}{2\epsilon}I+c_{v}I-\frac{1}{4\epsilon}I,$

which gives the result. ∎

The above lemma guarantees several nice properties of the gain function
$\bar{K}_{\epsilon}$, as shown in the following lemmas. In particular, it
ensures Lipschitz continuity of $K_{\epsilon}$ in the spatial argument with a
Lipschitz constant depending on the spectral gap of $p_{\epsilon}(x,dy)$.

######  Lemma 2.2.

Uniform boundedness of $\bar{K}_{\epsilon}(x,\mu)$ and
$\nabla_{x}\bar{K}_{\epsilon}(x,\mu)$ in $x$. Let $\mu$ be any log-concave
density as per Assumption 2.1 with $c_{v}>\frac{1}{4\epsilon}$and
$h\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$ a
$C^{1}$ and globally Lipschitz function. Then the following uniform bounds
hold

$\displaystyle\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(x,\mu)\right\rVert}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}}{2c_{g}}\left(\frac{4\epsilon c_{g}-1}{2\epsilon
c_{g}-1}\right)\quad\forall\enskip x\in\mathbb{R}^{d}$ (2.2)
$\displaystyle\mathinner{\\!\left\lVert\nabla_{x}\bar{K}_{\epsilon}(x,\mu)\right\rVert}_{F}$
$\displaystyle\leq\frac{d\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}}{\epsilon}\frac{1}{c_{g}^{3/2}}\left(\frac{4\epsilon
c_{g}-1}{2\epsilon c_{g}-1}\right),\quad\forall\enskip x\in\mathbb{R}^{d}$
(2.3)

where $c_{g}$ is the spectral gap of $p_{\epsilon}(x,y)$, as defined in (2.1)
and $\mathinner{\\!\left\lVert\cdot\right\rVert}_{F}$ indicates the Frobenius
norm.

###### Proof.

Consider $\phi_{\epsilon}$, the solution of (1.11). For small enough
$\epsilon$, the Neumann series expansion of (1.11) gives

$\displaystyle\phi_{\epsilon}^{\infty}(y)$
$\displaystyle=\epsilon(h(y)-\hat{h}_{\epsilon})+\epsilon\sum_{n=1}^{\infty}\int_{E^{n}}(h(y_{n})-\hat{h}_{\epsilon})\prod_{k=1}^{n}p_{\epsilon}(y_{k-1},y_{k})dy_{1}dy_{2}\cdots
dy_{n}$ (2.4a)
$\displaystyle=\epsilon\sum_{n=0}^{\infty}T_{\epsilon}^{n}(h(y)-\hat{h}_{\epsilon}),$
(2.4b)

where $y_{0}=y$ and $\int_{E^{n}}$ indicates $n$ iterated integrals. The
Markov operator $T_{\epsilon}h(y)=\int h(z)p_{\epsilon}(y,z)dz$ is globally
Lipschitz in $y$ since for any $v\in\mathbb{R}^{d}$

$\displaystyle\langle\nabla_{y}T_{\epsilon}h(y),\nabla_{y}T_{\epsilon}h(y)\rangle$
$\displaystyle=\frac{1}{2\epsilon}\int
h(z)\langle\nabla_{y}T_{\epsilon}h(y),z-p_{\epsilon}*z(y)\rangle
p_{\epsilon}(y,z)dz$ (2.5a)
$\displaystyle=\frac{1}{2\epsilon}\int(h(z)-\hat{h}_{p})\langle\nabla_{y}T_{\epsilon}h(y),z-p_{\epsilon}*z(y)\rangle
p_{\epsilon}(y,z)dz$ (2.5b) $\displaystyle\leq\frac{1}{2\epsilon
c_{g}}\left(\int\mathinner{\\!\left\lVert\nabla
h\right\rVert}^{2}p_{\epsilon}(y,z)dz\right)^{1/2}\mathinner{\\!\left\lVert\nabla_{y}T_{\epsilon}h(y)\right\rVert},$
(2.5c)

where the last inequality arises from Lemma 2.1 and a direct application of
Theorem 1.1 in [CCeL13] (see also Lemma G.1). This implies that

$\displaystyle\mathinner{\\!\left\lVert\nabla_{y}T_{\epsilon}h(y)\right\rVert}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}}{2\epsilon c_{g}},$

which combined with (2.4b) implies that $\phi_{\epsilon}^{\infty}$ is globally
Lipschitz in the spatial argument, when $\mu$ satisfies Assumption 2.1 and $h$
is globally Lipschitz, since

$\displaystyle\mathinner{\\!\left\lVert\phi_{\epsilon}^{\infty}(x)-\phi_{\epsilon}^{\infty}(y)\right\rVert}$
$\displaystyle\leq\epsilon\sum_{n=0}^{\infty}\mathinner{\\!\left\lVert
T^{n}_{\epsilon}h(x)-T^{n}_{\epsilon}h(y)\right\rVert}$ (2.6a)
$\displaystyle\leq\epsilon\sum_{n=0}^{\infty}\left(\frac{1}{2\epsilon
c_{g}}\right)^{n}\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}\mathinner{\\!\left\lVert x-y\right\rVert}$ (2.6b)
$\displaystyle=\left(\frac{\epsilon\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}}{1-\frac{1}{2\epsilon
c_{g}}}\right)\mathinner{\\!\left\lVert x-y\right\rVert},$ (2.6c)

which holds so long as $2\epsilon c_{g}>1$, which is guaranteed by the
assumption on $c_{v}$. By the same calculations as in (2.5c), we have

$\displaystyle\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(x,\mu)\right\rVert}^{2}$
$\displaystyle=\frac{1}{2\epsilon}\int(\mathcal{R}_{\epsilon}(y)-p_{\epsilon}*\mathcal{R}_{\epsilon}(x))\langle\bar{K}_{\epsilon}(x,\mu),y-p_{\epsilon}*y(x)\rangle
p_{\epsilon}(x,y)dy$ $\displaystyle\leq\frac{1}{2\epsilon
c_{g}}\left(\int\mathinner{\\!\left\lVert\nabla\mathcal{R}_{\epsilon}\right\rVert}^{2}p_{\epsilon}(x,y)dy\right)^{1/2}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(x,\mu)\right\rVert}.$

Using (2.6c), we have that

$\displaystyle\mathinner{\\!\left\lVert\nabla\mathcal{R}_{\epsilon}\right\rVert}_{\infty}\leq\epsilon\mathinner{\\!\left\lVert\nabla
h\right\rVert}_{\infty}\left(\frac{4\epsilon c_{g}-1}{2\epsilon
c_{g}-1}\right),$ (2.7)

which then gives the first result (2.2). In order to prove the second result,
start with

$\displaystyle\nabla_{x}\bar{K}_{\epsilon}(x,\mu)=-\frac{1}{4\epsilon^{2}}\underbrace{\int(\mathcal{R}_{\epsilon}(y)-p_{\epsilon}*\mathcal{R}_{\epsilon}(x))(y-p_{\epsilon}*y(x))(y-p_{\epsilon}*y(x))^{T}p_{\epsilon}(x,y)dy.}_{=\mathrel{\mathop{\mathchar
58\relax}}A}$

The Frobenius norm of $\nabla_{x}\bar{K}_{\epsilon}$ satisfies the inequality

$\displaystyle\mathinner{\\!\left\lVert\nabla_{x}\bar{K}_{\epsilon}(x,\mu)\right\rVert}_{F}\leq\frac{1}{4\epsilon^{2}}\left(\sum_{j}\sum_{i}A_{ij}^{2}\right)^{1/2}.$
(2.8)

A uniform bound on the entries $A_{ij}$ can be obtained using similar
calculations as above involving the Cauchy-Schwarz inequality and Lemmas 2.1
and G.1. More specifically,

$\displaystyle\mathinner{\\!\left\lVert
A_{ij}\right\rVert}=\mathinner{\\!\left\lVert\int(\mathcal{R}_{\epsilon}(y)-p_{\epsilon}*\mathcal{R}_{\epsilon}(x))(y-p_{\epsilon}*y(x))_{i}(y-p_{\epsilon}*y(x))_{j}p_{\epsilon}(x,y)dy\right\rVert}$
$\displaystyle\leq\left(\int(\mathcal{R}_{\epsilon}(y)-p_{\epsilon}*\mathcal{R}_{\epsilon}(x))^{2}p_{\epsilon}(x,y)dy\right)^{1/2}$
$\displaystyle\times\left(\int(y-p_{\epsilon}*y(x))_{i}^{2}(y-p_{\epsilon}*y(x))_{j}^{2}p_{\epsilon}(x,y)dy\right)^{1/2}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\nabla\mathcal{R}_{\epsilon}\right\rVert}_{\infty}}{\sqrt{c_{g}}}\left(\int(y-p_{\epsilon}*y(x))_{i}^{2}(y-p_{\epsilon}*y(x))_{j}^{2}p_{\epsilon}(x,y)dy\right)^{1/2}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\nabla\mathcal{R}_{\epsilon}\right\rVert}_{\infty}}{\sqrt{c_{g}}}\left(\int(y-p_{\epsilon}*y(x))_{i}^{4}p_{\epsilon}(x,y)dy\right)^{1/4}\left(\int(y-p_{\epsilon}*y(x))_{j}^{4}p_{\epsilon}(x,y)dy\right)^{1/4}.$

To bound the remaining terms, first let $f(x,y)\mathrel{\mathop{\mathchar
58\relax}}=(y-p_{\epsilon}*y(x))^{2}$ and $f_{i}$ refer to the $i$th entry of
the vector valued function $f$, so that

$\displaystyle\int(y-p_{\epsilon}*y(x))_{i}^{4}p_{\epsilon}(x,y)dy$
$\displaystyle=\int(f_{i}(x,y)-p_{\epsilon}*f_{i}(x)+p_{\epsilon}*f_{i}(x))^{2}p_{\epsilon}(x,y)dy$
$\displaystyle\leq
2\int(f_{i}(x,y)-p_{\epsilon}*f_{i}(x))^{2}p_{\epsilon}(x,y)dy+2\left(p_{\epsilon}*f_{i}(x)\right)^{2}.$

Applying Lemmas 2.1 and G.1 to the two terms in the last equality yields

$\displaystyle\int(y-p_{\epsilon}*y(x))_{i}^{4}p_{\epsilon}(x,y)dy\leq\frac{8}{c_{g}^{2}}+\frac{2}{c_{g}^{2}}=\frac{10}{c_{g}^{2}}.$

Inserting the above estimate into the upper bound for
$\mathinner{\\!\left\lVert A_{ij}\right\rVert}$ and combining with (2.8) and
(2.7) yields the second result (2.3). ∎

##  3 Well-posedness for the $N$-approximate process

Let ${\bf X}_{t}=[X_{t}^{1};X_{t}^{2};...;X_{t}^{N}]\\\
\in\mathbb{R}^{dN\times 1}$ be the vector of particles, and also define ${\bf
V}_{t}=[V_{t}^{1};V_{t}^{2};...;V_{t}^{N}]\in\mathbb{R}^{(dN+1)\times 1}$. We
can rewrite the $N$-FPF in the following form

$\displaystyle d{\bf X}_{t}=B({\bf X}_{t})dt+I_{dN}d{\bf
V}_{t}+\mathcal{K}_{\epsilon}({\bf X}_{t})dZ_{t},$ (3.1)

where $I_{dN}$ indicates the identity matrix of size $dN$,
$B\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{dN}\rightarrow\mathbb{R}^{dN}$ and
$\mathcal{K}_{\epsilon}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{dN}\rightarrow\mathbb{R}^{dN}$ are measurable functions
given by

$\displaystyle B({\bf X}_{t})\mathrel{\mathop{\mathchar
58\relax}}=\begin{bmatrix}b(X^{1}_{t},L_{N}({\bf X}_{t}))\\\
b(X_{t}^{2},L_{N}({\bf X}_{t}))\\\ \vdots\\\ b(X_{t}^{N},L_{N}({\bf
X}_{t}))\end{bmatrix};\quad\mathcal{K}_{\epsilon}({\bf
X}_{t})\mathrel{\mathop{\mathchar
58\relax}}=\begin{bmatrix}K_{\epsilon}(X^{1}_{t},L_{N}({\bf X}_{t}))\\\
K_{\epsilon}(X_{t}^{2},L_{N}({\bf X}_{t}))\\\ \vdots\\\
K_{\epsilon}(X_{t}^{N},L_{N}({\bf X}_{t})),\end{bmatrix}$

and

$\displaystyle b(X_{t}^{i},L_{N}({\bf X}_{t}))$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{M}(X_{t}^{i})-\frac{1}{2}K_{\epsilon}(X_{t}^{i},L_{N}({\bf
X}_{t}))(h(X_{t}^{i})+\hat{h}_{t}^{N}),$

where $\rho_{t}^{N}=L_{N}({\bf X}_{t})$ and $L_{N}\mathrel{\mathop{\mathchar
58\relax}}(\mathbb{R}^{d})^{N}\rightarrow\mathcal{P}(\mathbb{R}^{d})$ is the
empirical measure map given by

$\displaystyle
L_{N}(x_{1},x_{2},\cdots,x_{N})=\frac{1}{N}\sum_{k=1}^{N}\delta_{x_{k}}.$

It then follows from standard Ito theory that if $B,\mathcal{K}_{\epsilon}$
are $C^{2}$ (and therefore locally Lipschitz) and of linear growth, that the
SDE (3.1) has a unique strong solution. The proof of Theorem 1.1 follows
immediately from the following lemma and the fact that $h$ is assumed to be
uniformly bounded and $\mathcal{M}$ globally Lipschitz. As will be clear in
the proof of Lemma 3.1, this well-posedness only holds for finite $N$.
Furthermore, we restrict our attention to cases where the fixed point equation
is only solved for $L$ iterates, whose solution we denote by $\Phi^{(L)}$.
This is due to the difficulty in obtaining a control on the mixing rate of the
Markov chain uniformly in the particles without apriori conditions, see e.g.
the stopping time criterion in Section 4. In the following, we drop the time
subscript as the results hold for all $t\geq 0$.

######  Lemma 3.1.

Assume that $h\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$ is globally Lipschitz and
uniformly bounded and the fixed point equation (1.14) solved for $L$ iterates.
Then the function $\mathcal{K}_{\epsilon}(\bf X)$ satisfies

$\displaystyle\mathinner{\\!\left\lVert\mathcal{K}_{\epsilon}(\bf
X)\right\rVert}<C_{k}(1+\mathinner{\\!\left\lVert{\bf X}\right\rVert}),$ (3.2)

where

$\displaystyle
C_{k}=N^{6}d^{3/2}\sqrt{\epsilon}(1\vee\epsilon)(L_{h}+L\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}).$

###### Proof.

Recall the definition of $K_{\epsilon}$ from (1.18b). We first show that
$s_{ij}$ is uniformly bounded for all $i,j$. Specifically,

$\displaystyle\mathinner{\\!\left\lVert s_{ij}\right\rVert}$
$\displaystyle=\left|\frac{1}{2\epsilon}T_{ij}\left(\sum_{k=1}^{N}T_{ik}(r_{j}-r_{k})\right)\right|$
$\displaystyle\leq\frac{1}{2\epsilon}\left(\sum_{k=1}^{N}T_{ij}T_{ik}|r_{j}-r_{i}|+T_{ij}T_{ik}|r_{i}-r_{k}|\right)$
$\displaystyle\leq\frac{1}{2\epsilon}\left(T_{ij}|r_{j}-r_{i}|+\sum_{k=1}^{N}T_{ik}|r_{i}-r_{k}|\right).$

For the terms in the summand, notice that

$\displaystyle T_{ik}|r_{i}-r_{k}|$ $\displaystyle\leq
T_{ik}|\Phi_{i}^{(L)}-\Phi_{k}^{(L)}|+\epsilon T_{ik}L_{h}|X^{i}-X^{k}|.$
(3.3)

It follows from Jensen’s inequality that $T_{ik}\leq
Ng_{\epsilon}(X^{i},X^{k})$ and therefore

$\displaystyle T_{ik}|X^{i}-X^{k}|\leq
N2\sqrt{d\epsilon}\exp\left(-\frac{1}{2}\right).$ (3.4)

Additionally, we have that

$\displaystyle\mathinner{\\!\left\lVert\Phi_{k}^{(L)}-\Phi_{i}^{(L)}\right\rVert}$
$\displaystyle\leq\epsilon L_{h}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}+\epsilon\mathinner{\\!\left\lVert\sum_{n=1}^{L}[T^{n}{\bf
h}]_{k}-[T^{n}{\bf h}]_{i}\right\rVert}$ $\displaystyle\leq\epsilon
L_{h}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}LN\sum_{j}\mathinner{\\!\left\lVert
T_{kj}-T_{lm}\right\rVert}.$

The summand can be analysed starting from (1.12b) and recalling the definition
of $s_{\epsilon}^{i}$ in (1.13) as

$\displaystyle\mathinner{\\!\left\lVert T_{kj}-T_{ij}\right\rVert}$
$\displaystyle\leq\frac{1}{s_{\epsilon}^{k}}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{j})-\tilde{q}_{\epsilon}(X^{i},X^{j})\right\rVert}+\tilde{q}_{\epsilon}(X^{i},X^{j})\frac{1}{s_{\epsilon}^{k}s_{\epsilon}^{i}}\mathinner{\\!\left\lVert
s_{\epsilon}^{i}-s_{\epsilon}^{k}\right\rVert}$
$\displaystyle<\frac{1}{s_{\epsilon}^{k}}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{j})-\tilde{q}_{\epsilon}(X^{i},X^{j})\right\rVert}+\frac{1}{s_{\epsilon}^{k}s_{\epsilon}^{i}}\mathinner{\\!\left\lVert
s_{\epsilon}^{i}-s_{\epsilon}^{k}\right\rVert}$
$\displaystyle<\sqrt{N}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{j})-\tilde{q}_{\epsilon}(X^{i},X^{j})\right\rVert}+N\mathinner{\\!\left\lVert
s_{\epsilon}^{i}-s_{\epsilon}^{k}\right\rVert}$
$\displaystyle<\sqrt{N}\mathinner{\\!\left\lVert
g_{\epsilon}(X^{k},X^{j})-g_{\epsilon}(X^{i},X^{j})\right\rVert}+N\sum_{l}\mathinner{\\!\left\lVert
g_{\epsilon}(X^{k},X^{l})-g_{\epsilon}(X^{i},X^{l})\right\rVert}$
$\displaystyle<2N^{2}K_{g}\mathinner{\\!\left\lVert X^{k}-X^{i}\right\rVert},$

where for the last inequality, we have used (1.1). Combining all gives

$\displaystyle\mathinner{\\!\left\lVert\Phi_{k}^{(L)}-\Phi_{i}^{(L)}\right\rVert}$
$\displaystyle<D_{\phi}\mathinner{\\!\left\lVert X^{k}-X^{i}\right\rVert},$

where $D_{\phi}\mathrel{\mathop{\mathchar
58\relax}}=\epsilon\left(L_{h}+4LN^{3}\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}d\epsilon\exp(-1)\right)$. Substituting the above into
(3.3) gives

$\displaystyle T_{ik}|r_{i}-r_{k}|$ $\displaystyle\lesssim
N^{4}(d\epsilon)^{3/2}(1\vee\epsilon)(L_{h}+L\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}),$

so that for all $i,j$,

$\displaystyle\mathinner{\\!\left\lVert
s_{ij}\right\rVert}\lesssim\underbrace{N^{5}d^{3/2}\sqrt{\epsilon}(1\vee\epsilon)(L_{h}+L\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}).}_{=\mathrel{\mathop{\mathchar 58\relax}}C_{s}}$

The result then immediately follows from the fact that

$\displaystyle\mathinner{\\!\left\lVert\mathcal{K}_{\epsilon}({\bf
X})\right\rVert}<NC_{s}\mathinner{\\!\left\lVert\bf X\right\rVert}.$

∎

##  4 Propagation of chaos

Here we utilise the conditional propagation of chaos technique to establish
the mean field limit of (1.19). The standard coupling technique involves
considering $N$ independent copies of (1.20),
$\\{\bar{X}_{t}^{i}\\}_{i\in\\{1,2,\cdots N\\}}$ with the same Brownian
motions $\\{V^{i}_{t}\\}$ and iid initial conditions $\\{X_{0}^{i}\\}$, i.e.

$\displaystyle
d\bar{X}_{t}^{i}=\mathcal{M}(\bar{X}_{t}^{i})dt+dV_{t}^{i}+\bar{K_{\epsilon}}(\bar{X}_{t}^{i},\bar{\rho}_{t})\left(dZ_{t}-\frac{1}{2}(h(\bar{X}_{t}^{i})+\hat{\bar{h}}_{t})dt\right);\quad\bar{X}^{i}_{0}=X_{0}^{i},$

where we use $\bar{\rho}_{t}$ with a slight abuse of notation. We also use
$\rho_{t}^{N}$ to denote the empirical measure of the particles. The
conditional case simply involves conditioning on the common noise, in this
case, the observation process $Z$.

As mentioned in Section 1.3, we provide a result up to a stopping time only
due to the difficulties in establishing convergence of
$\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})$ to
$K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})$. The following stopping time arguments
effectively restrict the results to compact domains. Furthermore, this ensures
that there exists a unique solution $\phi_{\epsilon}$ as established formally
in [TMM20] for a measure $\mu$ with bounded support (and satisfying further
conditions). Well-posedness of the finite $N$ process follows from the same
arguments as in the proof of Theorem 1.1; the bounded $h$ assumption is easily
removed when considering a time interval up to the stopping time. The overall
goal is to first establish a uniform in $N$ control on the term
$\mathinner{\\!\left\lVert
K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})-\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})\right\rVert}$
which is facilitated by several intermediate results of a similar nature
involving the solution of the fixed point equation (Lemma 4.2 and 4.3) and on
the transition matrix (Lemma 4.1). The final crucial ingredient is a strong
law of large numbers type result, given in Lemma 4.5.

More specifically, consider the stopping times

$\displaystyle\tau_{\delta}^{N}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\inf\\{t\geq 0\mathrel{\mathop{\mathchar 58\relax}}\min_{1\leq
i\leq
N}\frac{1}{N}\sum_{k}\tilde{q}_{\epsilon}(X^{i}_{t},X^{k}_{t})<\delta\\}$
$\displaystyle\bar{\tau}_{\delta}^{N}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=\inf\\{t\geq
0\mathrel{\mathop{\mathchar 58\relax}}\min_{1\leq i\leq
N}\frac{1}{N}\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i}_{t},\bar{X}^{k}_{t})<\delta\\}$
$\displaystyle\bar{\tau}_{\delta}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\inf\\{t\geq 0\mathrel{\mathop{\mathchar 58\relax}}\min_{1\leq
i\leq N}\int g_{\epsilon}(\bar{X}^{i}_{t},y)\rho_{t}(y)dy<\delta\\}$

where $0<\delta<1$. Furthermore, define

$\displaystyle\zeta_{\delta}^{N}\mathrel{\mathop{\mathchar
58\relax}}=T\wedge\tau_{\delta}^{N}\wedge\bar{\tau}_{\delta}^{N}\wedge\bar{\tau}_{\delta}$

###  4.1 Intermediate results

Before proving Theorem 1.2, we establish some important intermediate results.
All results here apply for $t<\zeta_{\delta}^{N}$. Clearly we have

$\displaystyle s_{\epsilon}^{i}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{k=1}^{N}\tilde{q}_{\epsilon}(X^{i}_{t},X^{k}_{t})\geq\delta
N\quad\forall\enskip i.$ (4.1)

Additionally, it holds that

$\displaystyle\tilde{q}_{\epsilon}(X_{t}^{i},X_{t}^{k})<g_{\epsilon}(X_{t}^{i},X_{t}^{k}).$

This leads to the following upper and lower bounds (alternative
characterisations are possible)

$\displaystyle\frac{1}{N^{3/2}}g_{\epsilon}(X_{t}^{i},X_{t}^{k})$
$\displaystyle<\frac{1}{N}\tilde{q}_{\epsilon}(X_{t}^{i},X_{t}^{k})<T_{ik}<\frac{1}{\delta
N}\tilde{q}_{\epsilon}(X_{t}^{i},X_{t}^{k})<\frac{1}{\delta
N}g_{\epsilon}(X_{t}^{i},X_{t}^{k}).$ (4.2)

Furthermore, we have using (4.1) and (4.2) that

$\displaystyle\tilde{q}_{\epsilon}(X^{j},X^{m})$
$\displaystyle\leq\frac{g_{\epsilon}(X^{j},X^{m})}{\sqrt{\delta
N}}\leq\frac{1}{\sqrt{\delta N}}$ (4.3) $\displaystyle T_{jm}$
$\displaystyle\leq\frac{1}{(\delta N)^{3/2}}\quad\forall\enskip j,m.$ (4.4)

The aforementioned results also hold for these quantities evaluated on
$\\{\bar{X}_{t}^{i}\\}_{i=1\mathrel{\mathop{\mathchar 58\relax}}N}$, e.g.
where $\bar{T}_{ik}$ is given by

$\displaystyle\bar{T}_{ik}\mathrel{\mathop{\mathchar
58\relax}}=\frac{\tilde{q}_{\epsilon}(\bar{X}_{t}^{i},\bar{X}_{t}^{k})}{\sum_{l}\tilde{q}_{\epsilon}(\bar{X}_{t}^{i},\bar{X}_{t}^{l})}.$

We first introduce the following condition on $N$ which aids in obtaining
estimates on the fixed point equation (1.15). Many of the estimates described
below are therefore valid for “large enough” $N$ depending on the choice of
$\delta$, specifically,

######  Assumption 4.1.

For a given $0<\delta<1$,

$\displaystyle N>\frac{1}{4\delta^{3}}.$

The first intermediate result is the following Lipschitz type property, its
proof can be found in Appendix B. Throughout this section, we use the notation
$a\lesssim b$ to mean $a\leq Cb$ where $C$ is a constant independent of
$\epsilon,d,N,\delta$.

######  Lemma 4.1.

It holds for $t<\zeta_{\delta}^{N}$, for all $i,k$ that

$\displaystyle\mathinner{\\!\left\lVert T_{ik}-\bar{T}_{ik}\right\rVert}$
$\displaystyle\lesssim
d\epsilon\left(\frac{1}{\delta^{3}N^{3/2}}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+\frac{1}{\delta^{2}N^{3/2}}\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+\frac{1}{\delta^{4}N^{2}}\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}\right)$ (4.5)

and

$\displaystyle\left(\sum_{k}\mathinner{\\!\left\lVert
T_{ik}-\bar{T}_{ik}\right\rVert}\right)^{2}\lesssim
d^{2}\epsilon^{2}\left(\frac{1}{\delta^{6}N}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}^{2}+\frac{1}{\delta^{8}N}\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}^{2}\right)$ (4.6)

as well as

$\displaystyle\left(\sum_{k}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle\lesssim((d\epsilon)^{3}\vee
K_{f}^{2})\left(\frac{1}{\delta^{6}N}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}^{2}+\frac{1}{\delta^{4}N}\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}^{2}\right),$ (4.7)

where $K_{f}$ is defined in (1.2).

The following lemma shows that the difference of any two components of the
fixed point equation (1.14) satisfy a kind of linear growth type property. The
proof can be found in Appendix C and makes use of a modification to the
standard arguments for establishing contraction in the total variation metric
for transition matrices with strictly positive entries. The modified argument
along with Condition (4.1) allows us to indirectly satisfy a Doeblin
minorization condition.

######  Lemma 4.2.

It holds for $t<\zeta_{\delta}^{N}$, with a fixed $0<\delta<1$, $h$ globally
Lipschitz and uniformly bounded, that for all $k,l$ and $N$ satisfying
condition (4.1) that

$\displaystyle\mathinner{\\!\left\lVert r_{k}-r_{l}\right\rVert}\lesssim
C_{\phi}\mathinner{\\!\left\lVert X_{t}^{k}-X_{t}^{l}\right\rVert},$ (4.8)

and

$\displaystyle\mathinner{\\!\left\lVert\bar{r}_{k}-\bar{r}_{l}\right\rVert}<C_{\phi}\mathinner{\\!\left\lVert\bar{X}_{t}^{k}-\bar{X}_{t}^{l}\right\rVert},$

where

$\displaystyle C_{\phi}(N)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\epsilon\left(L_{h}+\frac{{d\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}}}{2\delta^{3/2}\sqrt{N}-1}\right),$ (4.9)

and $\bar{r}_{k}$ is given by (1.17) evaluated on
$\\{\bar{X}_{t}^{i}\\}_{i=1\mathrel{\mathop{\mathchar 58\relax}}N}$.
Additionally, it holds that

$\displaystyle\mathinner{\\!\left\lVert r_{k}-r_{l}\right\rVert}\leq
C_{\gamma}$ (4.10)

where

$\displaystyle C_{\gamma}\mathrel{\mathop{\mathchar
58\relax}}=2\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\frac{5\delta^{3/2}\sqrt{N}-2}{2\delta^{3/2}\sqrt{N}-1}\right),$

and likewise for $\bar{r}_{k}-\bar{r}_{l}$.

The contraction arguments used in the proof of the previous lemma also allow
us to establish the following ‘Lipschitz’ type property on $r$, analogous to
Lemma 4.1. The proof can be found in Appendix D.

######  Lemma 4.3.

Assume the same conditions as Lemma 4.2. Then it holds that for all $i,k$ and
$N$ satisfying condition (4.1) that

$\displaystyle\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{i}+\bar{r}_{k}\right\rVert}$ $\displaystyle\lesssim
C_{r1}\left(\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}\right)+C_{r2}\frac{1}{N}\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}$ (4.11)

where

$\displaystyle C_{r1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\left(\epsilon L_{h}+d\epsilon^{2}\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\frac{1}{2\delta^{3/2}\sqrt{N}-1}\right)\frac{1}{\delta^{3/2}}\right)$
$\displaystyle C_{r2}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{\epsilon^{2}d(L_{h}\vee\mathinner{\\!\left\lVert
h\right\rVert}_{\infty})}{\delta^{6}}\left(\frac{4\delta^{3/2}\sqrt{N}-1}{2\delta^{3/2}\sqrt{N}-1}\right),$

from which it follows immediately that

$\displaystyle\sum_{k}\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{i}+\bar{r}_{k}\right\rVert}^{2}\lesssim
C_{r1}^{2}N\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}^{2}+(C_{r1}^{2}+C_{r2}^{2})\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert}^{2}.$ (4.12)

The following lemma establishes a couple of useful properties of the empirical
$K_{\epsilon}$, its proof is given in Appendix E.

######  Lemma 4.4.

Assume the same conditions as Lemma 4.2. Then for $N$ sufficiently large, i.e.
satisfying condition (4.1), we have the following bounds,

$\displaystyle\mathinner{\\!\left\lVert
K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right\rVert}$ $\displaystyle\lesssim
K_{b},$ (4.13)

where

$\displaystyle K_{b}=C_{\phi}d,$

and

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\mathinner{\\!\left\lVert
K_{\epsilon}(\bar{X}^{i}_{t},\bar{\rho}_{t}^{N})-K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right\rVert}^{2}$
$\displaystyle\lesssim C_{K}\frac{1}{N}\sum_{i=1}^{N}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}^{2},$

where

$\displaystyle C_{K}=\frac{C_{\gamma}^{2}((d\epsilon)^{3}\vee
K_{f}^{2})}{\delta^{8}\epsilon^{2}}+\frac{d(C_{r1}^{2}+C_{r2}^{2})}{\epsilon}.$

A consequence of the above Lemma is that
$\nabla_{x}K_{\epsilon}(x,\mu_{t}^{N})$ has a uniformly bounded derivative
when $t<\zeta_{\delta}^{N}$ where the bound is independent of $N$ since
$K_{\epsilon}(x,\mu_{t}^{N})$ is continuously differentiable in $x$. This will
be useful when analysing the stratonovich correction term not considered here.

The following lemma will be needed to obtain a LLN type bound and relies on
the tail decay of the kernel $g_{\epsilon}(x,y)$ to control the functions
$\phi$ and $X$ in $\bar{K}_{\epsilon}$. We make use of the following
assumptions

######  Assumption 4.2.

There exists a solution $\phi_{\epsilon}$ to the fixed point equation (1.11).

######  Assumption 4.3.

For $t<\zeta_{\delta}^{N}$, there exists a constant $\delta_{v}>0$ such that

$\displaystyle\int\mathinner{\\!\left\lVert
y\right\rVert}\nu_{\epsilon}(y)dy<\delta_{v}.$

where $\nu_{\epsilon}$ is defined in (1.10).

######  Lemma 4.5.

Assume the same conditions as Lemma 4.2. Then for $N$ sufficiently large, i.e.
satisfying condition (4.1) it holds that

$\displaystyle\mathbb{E}\left[\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})-K_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t}^{N})\right\rVert}^{2}\right]\leq\frac{\mathcal{C}}{N},$
(4.14)

where

$\displaystyle\mathcal{C}=\frac{((\mathinner{\\!\left\lVert\phi_{\epsilon}\right\rVert}_{\infty}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty})\delta_{v}+C_{\gamma}d(\epsilon\vee\sqrt{\epsilon}))^{2}}{\epsilon^{2}\delta^{4}}.$

The proof can be found in Appendix F.

###  4.2 Proof of Theorem 1.2

We are now ready to prove Theorem 1.2 related to convergence of the finite $N$
process to the mean field process up to a stopping time. For any $i$, define

$\displaystyle\mathcal{X}_{t}^{i}\mathrel{\mathop{\mathchar
58\relax}}=\bar{X}_{t}^{i}-X_{t}^{i}.$

We have

$\displaystyle
d\mathcal{X}_{t}^{i}=[\mathcal{M}(\bar{X}_{t}^{i})-\mathcal{M}(X_{t}^{i})]dt+\left[\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})-K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right]h(X_{t}^{*})dt$
$\displaystyle-\frac{1}{2}\left[\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})-K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right]\left[h(\bar{X}_{t}^{i})+\widehat{\bar{h}}_{t}\right]dt-\frac{1}{2}K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\left[h(\bar{X}_{t}^{i})-h(X_{t}^{i})\right]dt$
$\displaystyle-\frac{1}{2}K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\left[\widehat{\bar{h}}_{t}-\hat{h}_{t}^{N}\right]dt+[\bar{K}_{\epsilon}(\bar{X}_{t}^{i},\bar{\rho}_{t})-K_{\epsilon}(X_{t}^{i},\bar{\rho}_{t})]dW_{t}.$

Using (4.13) from Lemma 4.4, we have that

$\displaystyle\sup_{s\in[0,t\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{s}^{i}\right\rVert}^{2}\leq
L_{M}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{u}^{i}\right\rVert}^{2}ds$
$\displaystyle+\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(\bar{X}_{s}^{i},\bar{\rho}_{s})-K_{\epsilon}(X_{s}^{i},\rho_{s}^{N})\right\rVert}^{2}ds$
$\displaystyle+\frac{1}{4}L_{h}^{2}K_{b}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{u}^{i}\right\rVert}^{2}ds+\frac{1}{4}K_{b}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathinner{\\!\left\lVert\widehat{\bar{h}}_{s}-\hat{h}_{s}^{N}\right\rVert}^{2}ds$
$\displaystyle+\sup_{s\in[0,t\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\int_{0}^{s}\bar{K}_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u})-K_{\epsilon}(X_{u}^{i},\rho_{u}^{N})dW_{u}\right\rVert}^{2}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=I_{3}+I_{4}+I_{5}+I_{6}+I_{7},$

where $X_{t}^{*}$ denotes the references process. Starting with the term
$I_{7}$, it follows from Doob’s maximal inequality and Ito isometry that

$\displaystyle\mathbb{E}\left[I_{7}\right]\lesssim\mathbb{E}\left[\int_{0}^{t\wedge\zeta_{\delta}^{N}}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u})-K_{\epsilon}(X_{u}^{i},\rho_{u}^{N})\right\rVert}^{2}du\right].$

Furthermore, Lemma 4.4 and Lemma 4.5 yield

$\displaystyle\mathbb{E}\left[\int_{0}^{t\wedge\zeta_{\delta}^{N}}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u})-K_{\epsilon}(X_{u}^{i},\rho_{u}^{N})\right\rVert}^{2}du\right]$
$\displaystyle\leq\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathbb{E}\left[\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\bar{K}_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u})-K_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u}^{N})\right\rVert}^{2}\right]du$
$\displaystyle+\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathbb{E}\left[\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert{K}_{\epsilon}(\bar{X}_{u}^{i},\bar{\rho}_{u}^{N})-K_{\epsilon}(X_{u}^{i},\rho_{u}^{N})\right\rVert}^{2}\right]ds$
$\displaystyle\leq\frac{\mathcal{C}}{N}(t\wedge\zeta_{\delta}^{N})+C_{K}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathbb{E}\left[\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{u}^{i}\right\rVert}^{2}\right]ds,$

which can also be used to bound $I_{4}$. To analyse $I_{6}$, notice that

$\displaystyle\mathinner{\\!\left\lVert\widehat{\bar{h}}_{s}-\hat{h}_{s}^{N}\right\rVert}^{2}$
$\displaystyle\leq
2\mathinner{\\!\left\lVert\widehat{\bar{h}}_{s}-\frac{1}{N}\sum_{i}h(\bar{X}_{s}^{i})\right\rVert}^{2}+2\mathinner{\\!\left\lVert\frac{1}{N}\sum_{i}h(\bar{X}_{s}^{i})-h(X_{s}^{i})\right\rVert}^{2}$
$\displaystyle\leq
2\mathinner{\\!\left\lVert\widehat{\bar{h}}_{s}-\frac{1}{N}\sum_{j}h(\bar{X}_{s}^{i})\right\rVert}^{2}+2\frac{L_{h}^{2}}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{s}^{i}\right\rVert}^{2},$

so that from the strong law of large numbers, we have

$\displaystyle\mathbb{E}\left[\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathinner{\\!\left\lVert\widehat{\bar{h}}_{s}-\hat{h}_{s}^{N}\right\rVert}^{2}ds\right]$
$\displaystyle\lesssim\frac{1}{N}\int_{0}^{t\wedge\zeta_{\delta}^{N}}Var(h(\bar{X}_{s}))ds$
$\displaystyle+L_{h}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathbb{E}\left[\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{u}^{i}\right\rVert}^{2}\right]ds$
$\displaystyle\lesssim\frac{\mathinner{\\!\left\lVert
h\right\rVert}^{2}_{\infty}(t\wedge\zeta_{\delta}^{N})}{N}$
$\displaystyle+L_{h}^{2}\int_{0}^{t\wedge\zeta_{\delta}^{N}}\mathbb{E}\left[\sup_{u\in[0,s\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{u}^{i}\right\rVert}^{2}\right]ds.$

Combining the above estimates along with Gronwall lemma gives

$\displaystyle\mathbb{E}\left[\sup_{s\in[0,t\wedge\zeta_{\delta}^{N}]}\frac{1}{N}\sum_{i}\mathinner{\\!\left\lVert\mathcal{X}_{s}^{i}\right\rVert}^{2}\right]\lesssim\frac{D_{9}\exp(D_{8}(t\wedge\zeta_{\delta}^{N}))}{N},$

where

$\displaystyle D_{8}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=(L_{M}^{2}+L_{h}^{2}K_{b}^{2}+\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}^{2}C_{K})$ $\displaystyle D_{9}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=(K_{b}^{2}+\mathcal{C})\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}^{2}(t\wedge\zeta_{\delta}^{N}).$

## Acknowledgments

We are grateful for constructive comments from anonymous referees that have
helped improve this article.

##  Appendix A Useful analytic estimates

The following lipschitz property of $g_{\epsilon}(x,y)$ will be used
extensively.

$\displaystyle\mathinner{\\!\left\lVert
g_{\epsilon}(x_{1},y_{1})-g_{\epsilon}(x_{2},y_{2})\right\rVert}\leq
K_{g}\mathinner{\\!\left\lVert x_{1}-y_{1}-x_{2}+y_{2}\right\rVert},$ (1.1)

with $K_{g}\mathrel{\mathop{\mathchar 58\relax}}=2d\epsilon\exp(-1)$.
Additionally, we have the following result.

######  Lemma A.1.

Consider $f\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ where
$f(z)=z\exp(-\frac{1}{4\epsilon}\mathinner{\\!\left\lVert z\right\rVert}^{2})$
for $\epsilon>0$. It holds that $f$ is globally lipschitz, i.e.

$\displaystyle\mathinner{\\!\left\lVert f(z)-f(y)\right\rVert}\leq
K_{f}\mathinner{\\!\left\lVert z-y\right\rVert},$

where

$\displaystyle K_{f}=\sqrt{\exp(-2)d^{2}+d(1-\exp(-2))}.$ (1.2)

###### Proof.

We start with
$f_{j}(z)=z_{j}\exp(-\frac{1}{4\epsilon}\mathinner{\\!\left\lVert
z\right\rVert}^{2})$ which is a scalar function. Then

$\displaystyle\mathinner{\\!\left\lVert\nabla_{j}f_{j}\right\rVert}$
$\displaystyle=\frac{1}{2\epsilon}\exp(-\frac{1}{4\epsilon}\mathinner{\\!\left\lVert
z\right\rVert}^{2})\mathinner{\\!\left\lVert 2\epsilon-z_{j}^{2}\right\rVert}$
$\displaystyle\leq\frac{1}{2\epsilon}\exp(-\frac{1}{4\epsilon}z_{j}^{2})\mathinner{\\!\left\lVert
2\epsilon-z_{j}^{2}\right\rVert}$ $\displaystyle\leq 1,$

and also

$\displaystyle\mathinner{\\!\left\lVert\nabla_{i}f_{j}\right\rVert}$
$\displaystyle=\frac{1}{2\epsilon}\mathinner{\\!\left\lVert
z_{j}z_{i}\right\rVert}\exp(-\frac{1}{4\epsilon}\mathinner{\\!\left\lVert
z\right\rVert}^{2}),\quad i\neq j$
$\displaystyle\leq\frac{1}{2\epsilon}\mathinner{\\!\left\lVert
z_{i}\right\rVert}\exp(-\frac{1}{4\epsilon}z_{i}^{2})\mathinner{\\!\left\lVert
z_{j}\right\rVert}\exp(-\frac{1}{4\epsilon}z_{j}^{2})$
$\displaystyle\leq\exp(-1).$

So we have shown that the individual entries of the vector function $f$ are
lipschitz, meaning

$\displaystyle\mathinner{\\!\left\lVert f(z)-f(y)\right\rVert}$
$\displaystyle=\left(\sum_{j}(f_{j}(y)-f_{j}(z))^{2}\right)^{1/2}\leq\left(\mathinner{\\!\left\lVert
y-z\right\rVert}^{2}\sum_{j}\left[(\nabla_{j}f_{j})^{2}+\sum_{i\neq
j}(\nabla_{i}f_{j})^{2}\right]\right)^{1/2}$
$\displaystyle\leq\sqrt{\exp(-2)d^{2}+d(1-\exp(-2))}\mathinner{\\!\left\lVert
y-z\right\rVert}.$

∎

##  Appendix B Proof of lemma 4.1

For notational ease, we drop the time subscript in the following which is held
fixed. We have that

$\displaystyle
T_{ik}-\bar{T}_{ik}=\frac{\tilde{q}_{\epsilon}(X^{i},X^{k})}{\sum_{l}\tilde{q}_{\epsilon}(X^{i},X^{l})}-\frac{\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}{\sum_{l}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{l})},$

and let us adopt the shorthand notation
$\bar{s}_{\epsilon}^{i}=\sum_{l}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{l}).$
Then

$\displaystyle\mathinner{\\!\left\lVert
T_{ik}-\bar{T}_{ik}\right\rVert}\leq\frac{\mathinner{\\!\left\lVert\bar{s}^{i}_{\epsilon}-s^{i}_{\epsilon}\right\rVert}}{s^{i}_{\epsilon}\bar{s}^{i}_{\epsilon}}\tilde{q}_{\epsilon}(X^{i},X^{k})+\frac{1}{\bar{s}_{\epsilon}^{i}}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{i},X^{k})-\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\right\rVert}.$

Similarly, we have using the shorthand notation
$u_{\epsilon}^{k}\mathrel{\mathop{\mathchar
58\relax}}=(\sum_{l}g_{\epsilon}(X^{k},X^{l}))^{1/2}$ that

$\displaystyle\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{i},X^{k})-\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\right\rVert}$
$\displaystyle=\mathinner{\\!\left\lVert\frac{g_{\epsilon}(X^{i},X^{k})}{u_{\epsilon}^{k}}-\frac{g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}{\bar{u}_{\epsilon}^{k}}\right\rVert}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\bar{u}^{k}_{\epsilon}-u^{k}_{\epsilon}\right\rVert}}{u^{k}_{\epsilon}\bar{u}^{k}_{\epsilon}}g_{\epsilon}(X^{i},X^{k})+\frac{1}{\bar{u}_{\epsilon}^{k}}\mathinner{\\!\left\lVert
g_{\epsilon}(X^{i},X^{k})-g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\right\rVert}$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\bar{u}^{k}_{\epsilon}-u^{k}_{\epsilon}\right\rVert}}{u^{k}_{\epsilon}\bar{u}^{k}_{\epsilon}}+\frac{1}{\bar{u}_{\epsilon}^{k}}K_{g}\left(\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}\right)$
$\displaystyle\leq\frac{\mathinner{\\!\left\lVert\bar{u}^{k}_{\epsilon}-u^{k}_{\epsilon}\right\rVert}}{\delta
N}+\frac{1}{\sqrt{\delta N}}K_{g}\left(\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}\right).$

In the above we have used (4.2) and (1.1). Now notice that

$\displaystyle
u_{\epsilon}^{k}=\sqrt{\sum_{l}(g_{2\epsilon}(X^{k},X^{l}))^{2}}=\mathinner{\\!\left\lVert
G_{2\epsilon}({\bf z})\right\rVert},$

where $G_{2\epsilon}({\bf z})\mathrel{\mathop{\mathchar
58\relax}}=\left[g_{2\epsilon}(z_{1}),g_{2\epsilon}(z_{2}),\cdots,g_{2\epsilon}(z_{N})\right]^{T}$
and $z_{l}\mathrel{\mathop{\mathchar 58\relax}}=X^{k}-X^{l}$. By the reverse
triangle inequality and (1.1), we have

$\displaystyle\mathinner{\\!\left\lVert
u_{\epsilon}^{k}-\bar{u}_{\epsilon}^{k}\right\rVert}$
$\displaystyle=\mathinner{\\!\left\lVert\mathinner{\\!\left\lVert
G_{2\epsilon}({\bf z})\right\rVert}-\mathinner{\\!\left\lVert
G_{2\epsilon}({\bf\bar{z}})\right\rVert}\right\rVert}$
$\displaystyle\leq\mathinner{\\!\left\lVert G_{2\epsilon}({\bf
z})-G_{2\epsilon}({\bf\bar{z}})\right\rVert}$ $\displaystyle\leq
2K_{g}\left(\sum_{l}\mathinner{\\!\left\lVert
X^{k}-X^{l}-(\bar{X}^{k}-\bar{X}^{l})\right\rVert}^{2}\right)^{1/2}$
$\displaystyle\leq 2\sqrt{2}K_{g}\left(\sqrt{N}\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+\left(\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}^{2}\right)^{1/2}\right).$

Combining gives

$\displaystyle\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{i},X^{k})-\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\right\rVert}$
$\displaystyle\lesssim\frac{K_{g}}{\sqrt{\delta N}}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+K_{g}\left(\frac{1}{\delta\sqrt{N}}+\frac{1}{\sqrt{\delta
N}}\right)\mathinner{\\!\left\lVert X^{k}-\bar{X}^{k}\right\rVert}$
$\displaystyle+\frac{K_{g}}{\delta N}\left(\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}^{2}\right)^{1/2},$

from which it follows that

$\displaystyle\mathinner{\\!\left\lVert\bar{s}^{i}_{\epsilon}-s^{i}_{\epsilon}\right\rVert}\lesssim\frac{K_{g}\sqrt{N}}{\sqrt{\delta}}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}$
$\displaystyle+K_{g}\left(\frac{1}{\delta\sqrt{N}}+\frac{1}{\sqrt{\delta
N}}\right)\sum_{l}\mathinner{\\!\left\lVert X^{l}-\bar{X}^{l}\right\rVert}$
$\displaystyle+\frac{K_{g}}{\delta}\left(\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}^{2}\right)^{1/2},$

and then using (4.3) and (4.1), we have that

$\displaystyle\mathinner{\\!\left\lVert
T_{ik}-\bar{T}_{ik}\right\rVert}\lesssim
d\epsilon\left(\frac{1}{\delta^{3}N^{3/2}}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}+\frac{1}{\delta^{2}N^{3/2}}\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+\frac{1}{\delta^{4}N^{2}}\sum_{l}\mathinner{\\!\left\lVert
X^{l}-\bar{X}^{l}\right\rVert}\right),$

which gives the first result (4.5). The result (4.6) follows in a
straightforward manner using Jensen’s inequality. The calculations to obtain
(4.7) are entirely analogous (making use of Lemma A.1 instead of (1.1)) and
are therefore omitted. ∎

##  Appendix C Proof of lemma 4.2

Recall (1.15). For $n=0$, we have trivially that

$\displaystyle\mathinner{\\!\left\lVert[T^{0}{\bf h}]_{k}-[T^{0}{\bf
h}]_{l}\right\rVert}\leq L_{h}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert},$ (3.1)

and also

$\displaystyle\mathinner{\\!\left\lVert[T^{0}{\bf h}]_{k}-[T^{0}{\bf
h}]_{l}\right\rVert}\leq 2\mathinner{\\!\left\lVert h\right\rVert}_{\infty},$
(3.2)

and for $n=1$ it holds that

$\displaystyle\mathinner{\\!\left\lVert[T{\bf h}]_{k}-[T{\bf
h}]_{l}\right\rVert}\leq\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\sum_{m}\mathinner{\\!\left\lVert
T_{km}-T_{lm}\right\rVert}.$

Now consider $n=2$, where it holds that

$\displaystyle\mathinner{\\!\left\lVert[T^{2}{\bf h}]_{k}-[T^{2}{\bf
h}]_{l}\right\rVert}\leq\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\sum_{m}\mathinner{\\!\left\lVert[T^{2}]_{km}-[T^{2}]_{lm}\right\rVert},$

and

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert[T^{2}]_{km}-[T^{2}]_{lm}\right\rVert}$
$\displaystyle=\sum_{m}\mathinner{\\!\left\lVert\sum_{j}(T_{kj}-T_{lj})T_{jm}\right\rVert}$
$\displaystyle=2\mathinner{\\!\left\lVert(v_{k}-v_{l})T\right\rVert}_{TV},$

where $v_{k}\mathrel{\mathop{\mathchar
58\relax}}=[T_{k1},T_{k2},\cdots,T_{kN}]$. Since $v_{k}$ and $v_{l}$ are
probability vectors and $T$ is a Markov matrix with strictly positive entries,
it holds that $T$ is a strict contraction in the total variation metric. By a
slight modification of standard arguments and making us of (4.4), we can
define a matrix $Q$ with entries

$\displaystyle Q_{jm}=\frac{\delta\sqrt{\delta N}}{1-\delta\sqrt{\delta
N}}\left(\frac{1}{(\delta N)^{3/2}}-T_{jm}\right),$

which is a stochastic matrix since all the entries are non-negative and
$\sum_{m}Q_{jm}=1$. Therefore, we can write $T_{jm}=\frac{1}{(\delta
N)^{3/2}}-\frac{1-\delta\sqrt{\delta N}}{\delta\sqrt{\delta N}}Q_{jm}$

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert\sum_{j}(v_{k}(j)-v_{l}(j))T_{jm}\right\rVert}$
$\displaystyle=\sum_{m}\mathinner{\\!\left\lVert\sum_{j}(v_{k}(j)-v_{l}(j))\left(\frac{1}{(\delta
N)^{3/2}}-\frac{1-\delta\sqrt{\delta N}}{\delta\sqrt{\delta
N}}Q_{jm}\right)\right\rVert}$ $\displaystyle\leq\frac{1}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert\sum_{j}(v_{k}(j)-v_{l}(j))\right\rVert}$
$\displaystyle+\sum_{m}\mathinner{\\!\left\lVert\sum_{j}\left((v_{l}(j)-v_{k}(j))\frac{1-\delta\sqrt{\delta
N}}{\delta\sqrt{\delta N}}Q_{jm}\right)\right\rVert}$
$\displaystyle=\frac{1-\delta\sqrt{\delta N}}{\delta\sqrt{\delta
N}}\sum_{j}\mathinner{\\!\left\lVert
v_{l}(j)-v_{k}(j)\right\rVert}\sum_{m}Q_{jm}$
$\displaystyle=\frac{1-\delta\sqrt{\delta N}}{\delta\sqrt{\delta
N}}2\mathinner{\\!\left\lVert v_{k}-v_{l}\right\rVert}_{TV}.$

That is, we have that

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert[T^{2}]_{km}-[T^{2}]_{lm}\right\rVert}$
$\displaystyle\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)\sum_{j}\mathinner{\\!\left\lVert
T_{kj}-T_{lj}\right\rVert}.$

Inserting the above along with (3.2) into (1.15) and induction gives

$\displaystyle\mathinner{\\!\left\lVert\Phi^{\infty}_{k}-\Phi^{\infty}_{l}\right\rVert}$
$\displaystyle=\epsilon\mathinner{\\!\left\lVert\sum_{n=0}^{\infty}[T^{n}{\bf
h}]_{k}-[T^{n}{\bf h}]_{l}\right\rVert}$ $\displaystyle\leq
2\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\sum_{j}\mathinner{\\!\left\lVert
T_{kj}-T_{lj}\right\rVert}\right)\sum_{n=1}^{\infty}\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-1}$
$\displaystyle=2\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\sum_{j}\mathinner{\\!\left\lVert
T_{kj}-T_{lj}\right\rVert}\right)\frac{1}{2-\frac{1}{\delta^{3/2}\sqrt{N}}},$

where the geometric series converges so long as
$\mathinner{\\!\left\lVert\frac{1}{\delta^{3/2}\sqrt{N}}-1\right\rVert}<1$,
which is satisfied under condition (4.1). The result (4.10) then follows
immediately. To obtain the result (4.8), use (4.3) to obtain

$\displaystyle\mathinner{\\!\left\lVert
T_{km}-T_{lm}\right\rVert}\leq\frac{1}{s^{k}_{\epsilon}}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{m})-\tilde{q}_{\epsilon}(X^{l},X^{m})\right\rVert}+\tilde{q}_{\epsilon}(X^{l},X^{m})\frac{1}{s_{\epsilon}^{k}s_{\epsilon}^{l}}\mathinner{\\!\left\lVert
s_{\epsilon}^{l}-s_{\epsilon}^{k}\right\rVert}$
$\displaystyle\leq\frac{1}{\delta
N}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{m})-\tilde{q}_{\epsilon}(X^{l},X^{m})\right\rVert}+\frac{1}{(\delta
N)^{5/2}}\mathinner{\\!\left\lVert
s_{\epsilon}^{l}-s_{\epsilon}^{k}\right\rVert}$
$\displaystyle\leq\frac{1}{\delta
N}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{m})-\tilde{q}_{\epsilon}(X^{l},X^{m})\right\rVert}+\frac{1}{(\delta
N)^{5/2}}\sum_{j}\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{j})-\tilde{q}_{\epsilon}(X^{l},X^{j})\right\rVert},$

and it holds that

$\displaystyle\mathinner{\\!\left\lVert\tilde{q}_{\epsilon}(X^{k},X^{m})-\tilde{q}_{\epsilon}(X^{l},X^{m})\right\rVert}$
$\displaystyle\leq\frac{1}{\sqrt{\sum_{j}g_{\epsilon}(X^{m},X^{j})}}\mathinner{\\!\left\lVert
g_{\epsilon}(X^{k},X^{m})-g_{\epsilon}(X^{l},X^{m})\right\rVert}$
$\displaystyle\leq\frac{1}{\sqrt{\delta N}}{K_{g}}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert},$

using (1.1). Combining gives

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert T_{km}-T_{lm}\right\rVert}$
$\displaystyle\leq\frac{{2K_{g}}}{\delta^{3/2}\sqrt{N}}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert}.$ (3.3)

Again, inserting the above along with (3.1) into (1.15) gives

$\displaystyle\mathinner{\\!\left\lVert\Phi^{\infty}_{k}-\Phi^{\infty}_{l}\right\rVert}$
$\displaystyle\leq\epsilon L_{h}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\frac{{2K_{g}}}{\delta^{3/2}\sqrt{N}}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert}\sum_{n=1}^{\infty}\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-1}$
$\displaystyle\leq\epsilon L_{h}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\frac{{4\exp(-1)d\epsilon}}{\delta^{3/2}\sqrt{N}}\mathinner{\\!\left\lVert
X^{k}-X^{l}\right\rVert}\frac{1}{2-\frac{1}{\delta^{3/2}\sqrt{N}}}.$

∎

##  Appendix D Proof of lemma 4.3

Here we will make use of the following simplifying notation

$\displaystyle\mathcal{T}_{i}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\sum_{l}\mathinner{\\!\left\lVert
T_{il}-\bar{T}_{il}\right\rVert},$ $\displaystyle\mathcal{G}_{k}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\sum_{l}T_{kl}\sum_{m}\mathinner{\\!\left\lVert
T_{lm}-\bar{T}_{lm}\right\rVert}.$

With a slight abuse of notaiton, we also use $\mathcal{G}_{k}$ to denote
$\sum_{l}\bar{T}_{kl}\sum_{m}\mathinner{\\!\left\lVert
T_{lm}-\bar{T}_{lm}\right\rVert}$. Now it holds that

$\displaystyle\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{i}+\bar{r}_{k}\right\rVert}\leq\epsilon\left(2L_{h}\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+2L_{h}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}\right)$
$\displaystyle+\epsilon\sum_{n=1}^{\infty}\mathinner{\\!\left\lVert\underbrace{[T^{n}({\bf
h}-\hat{h}{\bf 1})]_{k}-[T^{n}({\bf h}-\hat{h}{\bf
1})]_{i}-[\bar{T}^{n}({\bf\bar{h}}-\hat{\bar{h}}{\bf
1})]_{k}+[\bar{T}^{n}({\bf\bar{h}}-\hat{\bar{h}}{\bf 1})]_{i}}\right\rVert}.$

We denote the summand in the above by $\mathcal{S}_{r}^{n}$. Notice that the
summand can be written as

$\displaystyle\mathcal{S}_{r}^{n}$
$\displaystyle=\sum_{m}([T^{n}]_{km}-\pi_{m}-[\bar{T}^{n}]_{km}+\bar{\pi}_{m})h(X^{m})$
$\displaystyle-\sum_{m}([T^{n}]_{im}-\pi_{m}-[\bar{T}^{n}]_{im}+\bar{\pi}_{m})h(X^{m})$
$\displaystyle+\sum_{m}([\bar{T}^{n}]_{km}-\bar{\pi}_{m})(h(X^{m})-h(\bar{X}^{m}))-\sum_{m}([\bar{T}^{n}]_{im}-\bar{\pi}_{m})(h(X^{m})-h(\bar{X}^{m}))$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}S_{k}^{n}-S_{i}^{n}+I_{k}^{n}-I_{i}^{n}.$

Starting with $S_{k}$,

$\displaystyle S_{k}^{n}-S_{i}^{n}$
$\displaystyle=\sum_{m}h(X^{m})\sum_{l}([T^{n-1}]_{kl}-[\bar{T}^{n-1}]_{kl})T_{lm}+([\bar{T}^{n-1}]_{kl}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})$
$\displaystyle-([T^{n-1}]_{il}-[\bar{T}^{n-1}]_{il})T_{lm}-([\bar{T}^{n-1}]_{il}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm}),$

So that

$\displaystyle\mathinner{\\!\left\lVert S_{k}^{n}-S_{i}^{n}\right\rVert}$
$\displaystyle\leq\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\sum_{m}\mathinner{\\!\left\lVert\sum_{l}([T^{n-1}]_{kl}-[\bar{T}^{n-1}]_{kl})T_{lm}\right\rVert}\right.$
$\displaystyle\left.+\sum_{m}\mathinner{\\!\left\lVert\sum_{l}([T^{n-1}]_{il}-[\bar{T}^{n-1}]_{il})T_{lm}\right\rVert}\right.$
$\displaystyle\left.+\sum_{m}\mathinner{\\!\left\lVert\sum_{l}([\bar{T}^{n-1}]_{kl}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})-([\bar{T}^{n-1}]_{il}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})\right\rVert}\right)$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}(U_{k}^{n}+U_{i}^{n}+V_{ik}^{n}).$

In order to analyse the second term in the above, consider first the case
$n=1$. Then we have

$\displaystyle V_{ik}^{1}$ $\displaystyle\leq\mathcal{T}_{k}+\mathcal{T}_{i}.$

Then for $n=2$

$\displaystyle V_{ik}^{2}$
$\displaystyle\leq\sum_{m}\mathinner{\\!\left\lVert\sum_{l}\bar{T}_{kl}(T_{lm}-\bar{T}_{lm})-\bar{T}_{il}(T_{lm}-\bar{T}_{lm})\right\rVert}$
$\displaystyle\leq\mathcal{G}_{k}+\mathcal{G}_{i}.$

Then for $n=3$,

$\displaystyle V_{ik}^{3}$
$\displaystyle\leq\sum_{m}\sum_{l}\mathinner{\\!\left\lVert([\bar{T}^{2}]_{kl}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})\right\rVert}+\mathinner{\\!\left\lVert([\bar{T}^{2}]_{il}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})\right\rVert}$
$\displaystyle\leq\sum_{j}\left(\mathinner{\\!\left\lVert\bar{T}_{kj}-\bar{\pi}_{j}\right\rVert}+\mathinner{\\!\left\lVert\bar{T}_{ij}-\bar{\pi}_{j}\right\rVert}\right)\mathcal{G}_{j},$

and $n=4$,

$\displaystyle V_{ik}^{4}$
$\displaystyle\leq\sum_{m}\sum_{l}\mathinner{\\!\left\lVert([\bar{T}^{3}]_{kl}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})\right\rVert}+\mathinner{\\!\left\lVert([\bar{T}^{3}]_{il}-\bar{\pi}_{l})(T_{lm}-\bar{T}_{lm})\right\rVert}$
$\displaystyle\leq\sum_{j}\left(\mathinner{\\!\left\lVert[\bar{T}^{2}]_{kj}-\bar{\pi}_{j}\right\rVert}+\mathinner{\\!\left\lVert[\bar{T}^{2}]_{ij}-\bar{\pi}_{j}\right\rVert}\right)\mathcal{G}_{j}.$

Now note that for $n\geq 1$, the following contraction holds

$\displaystyle\sum_{j}\mathinner{\\!\left\lVert[\bar{T}^{n}]_{kj}-\bar{\pi}_{j}\right\rVert}$
$\displaystyle\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-1}\sum_{j}\mathinner{\\!\left\lVert
T_{kj}-\pi_{j}\right\rVert}$ (4.1a) $\displaystyle\leq
2\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-1},$ (4.1b)

by similar reasoning as in the proof of Lemma 4.2. So that by induction, for
$n\geq 3$,

$\displaystyle V_{ik}^{n}$
$\displaystyle\leq\sum_{j}\left(\mathinner{\\!\left\lVert[\bar{T}^{n-2}]_{kj}-\bar{\pi}_{j}\right\rVert}+\mathinner{\\!\left\lVert[\bar{T}^{n-2}]_{ij}-\bar{\pi}_{j}\right\rVert}\right)\mathcal{G}_{j}$
$\displaystyle\leq
4\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-3}{\mathcal{G}},$

where $\mathcal{G}\mathrel{\mathop{\mathchar
58\relax}}=\max_{j}\mathcal{G}_{j}$. Finally,

$\displaystyle\sum_{n=1}^{\infty}V_{ik}^{n}$
$\displaystyle\lesssim\mathcal{T}_{k}+\mathcal{T}_{i}+\mathcal{G}_{k}+\mathcal{G}_{i}+4{\mathcal{G}}\sum_{n=3}^{\infty}\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-3}$
$\displaystyle\lesssim\mathcal{T}_{k}+\mathcal{T}_{i}+\mathcal{G}_{k}+\mathcal{G}_{i}+4{\mathcal{G}}\frac{\delta^{3/2}\sqrt{N}}{2\delta^{3/2}\sqrt{N}-1}.$

In order to obtain bounds on $U_{k}^{n}$, first consider the term

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert[T^{n}]_{km}-[\bar{T}^{n}]_{km}\right\rVert},$

inductively. The case $n=1$ is simply equivalent to $\mathcal{T}_{k}$. For
$n=2$, we have

$\displaystyle\mathinner{\\!\left\lVert\sum_{m}\sum_{l}T_{kl}T_{lm}-\bar{T}_{kl}\bar{T}_{lm}\right\rVert}$
$\displaystyle\leq\sum_{m}\mathinner{\\!\left\lVert\sum_{l}T_{kl}-\bar{T}_{kl}T_{lm}\right\rVert}+\mathinner{\\!\left\lVert\sum_{l}\bar{T}_{kl}\sum_{m}(T_{lm}-\bar{T}_{lm})\right\rVert}$
$\displaystyle=\sum_{m}\mathinner{\\!\left\lVert\sum_{l}T_{kl}-\bar{T}_{kl}T_{lm}\right\rVert}$
$\displaystyle\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)\mathcal{T}_{k},$

where the last inequality follows from the same reasoning as Lemma 4.2. Then
for $n=3$, we have

$\displaystyle\mathinner{\\!\left\lVert\sum_{m}\sum_{l}[T^{2}]_{kl}T_{lm}-[\bar{T}^{2}]_{kl}\bar{T}_{lm}\right\rVert}$
$\displaystyle\leq\sum_{m}\sum_{l}\mathinner{\\!\left\lVert[T^{2}]_{kl}-[\bar{T}^{2}]_{kl}\right\rVert}T_{lm}$
$\displaystyle+\mathinner{\\!\left\lVert\sum_{l}[\bar{T}^{2}]_{kl}\sum_{m}(T_{lm}-\bar{T}_{lm})\right\rVert}$
$\displaystyle\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)\sum_{l}\mathinner{\\!\left\lVert[T^{2}]_{kl}-[\bar{T}^{2}]_{kl}\right\rVert}$
$\displaystyle\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{2}\mathcal{T}_{k}.$

Again, by induction for $n\geq 1$,

$\displaystyle\sum_{m}\mathinner{\\!\left\lVert[T^{n}]_{km}-[\bar{T}^{n}]_{km}\right\rVert}\leq\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-1}\mathcal{T}_{k}.$

Turning to $U_{k}^{n}$, for $n\geq 2$ (clearly $U_{k}^{1}=0$),

$\displaystyle
U_{k}^{n}\leq\sum_{l}\mathinner{\\!\left\lVert[T^{n-1}]_{kl}-[\bar{T}^{n-1}]_{kl}\right\rVert}$

so that

$\displaystyle\sum_{n=1}^{\infty}U_{k}^{n}$
$\displaystyle\leq\sum_{n=1}^{\infty}\sum_{l}\mathinner{\\!\left\lVert[T^{n}]_{kl}-[\bar{T}^{n}]_{kl}\right\rVert}$
$\displaystyle\leq\sum_{l}\mathinner{\\!\left\lVert
T_{kl}-\bar{T}_{kl}\right\rVert}\sum_{n=0}^{\infty}\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n}$
$\displaystyle\leq\mathcal{T}_{k}\frac{\delta^{3/2}\sqrt{N}}{2\delta^{3/2}\sqrt{N}-1}.$

The analogous result holds for $U_{i}^{n}$. Combining all gives

$\displaystyle\sum_{n=1}^{\infty}\mathinner{\\!\left\lVert
S_{k}^{n}-S_{i}^{n}\right\rVert}$ $\displaystyle\leq\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left((\mathcal{T}_{k}+\mathcal{T}_{i}+4{\mathcal{G}})\frac{\delta^{3/2}\sqrt{N}}{2\delta^{3/2}\sqrt{N}-1}+\mathcal{T}_{k}+\mathcal{T}_{i}+\mathcal{G}_{k}+\mathcal{G}_{i}\right).$

The analysis for $I_{k}^{n}$ and $I_{i}^{n}$ follows in a similar fashion,
starting with $n=1$,

$\displaystyle\mathinner{\\!\left\lVert I_{k}^{1}-I_{i}^{1}\right\rVert}$
$\displaystyle\leq\mathinner{\\!\left\lVert\sum_{m}\bar{T}_{km}(h(X^{m})-h(\bar{X}^{m}))-\bar{T}_{im}(h(X^{m})-h(\bar{X}^{m}))\right\rVert}$
$\displaystyle\leq\frac{2L_{h}}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert X^{m}-\bar{X}^{m}\right\rVert},$

and for $n=2$,

$\displaystyle\mathinner{\\!\left\lVert I_{k}^{2}-I_{i}^{2}\right\rVert}$
$\displaystyle\leq\frac{L_{h}}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert}\left(\sum_{l}\mathinner{\\!\left\lVert\bar{T}_{kl}-\bar{\pi}_{l}\right\rVert}+\sum_{l}\mathinner{\\!\left\lVert\bar{T}_{il}-\bar{\pi}_{l}\right\rVert}\right)$
$\displaystyle\leq\frac{4L_{h}}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert X^{m}-\bar{X}^{m}\right\rVert},$

and for $n=3$,

$\displaystyle\mathinner{\\!\left\lVert I_{k}^{3}-I_{i}^{3}\right\rVert}$
$\displaystyle\leq\frac{L_{h}}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert}\left(\sum_{l}\mathinner{\\!\left\lVert[\bar{T}^{2}]_{kl}-\bar{\pi}_{l}\right\rVert}+\sum_{l}\mathinner{\\!\left\lVert[\bar{T}^{2}]_{il}-\bar{\pi}_{l}\right\rVert}\right),$

so that by induction, for $n\geq 2$

$\displaystyle\mathinner{\\!\left\lVert I_{k}^{n}-I_{i}^{n}\right\rVert}$
$\displaystyle\leq\frac{L_{h}}{(\delta
N)^{3/2}}\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert}\left(\sum_{l}\mathinner{\\!\left\lVert[\bar{T}^{n-1}]_{kl}-\bar{\pi}_{l}\right\rVert}+\sum_{l}\mathinner{\\!\left\lVert[\bar{T}^{n-1}]_{il}-\bar{\pi}_{l}\right\rVert}\right)$
$\displaystyle\leq\frac{4L_{h}}{(\delta
N)^{3/2}}\left(\frac{1}{\delta^{3/2}\sqrt{N}}-1\right)^{n-2}\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert},$

using (4.1b). Combining all, we have

$\displaystyle\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{i}+\bar{r}_{k}\right\rVert}$
$\displaystyle\leq\epsilon\left(2L_{h}\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+2L_{h}\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}\right)+\epsilon\sum_{n=1}^{\infty}\mathinner{\\!\left\lVert
S_{k}^{n}-S_{i}^{n}\right\rVert}$
$\displaystyle+\epsilon\sum_{n=1}^{\infty}\mathinner{\\!\left\lVert
I_{k}^{n}-I_{n}^{i}\right\rVert}$ $\displaystyle\lesssim\epsilon
L_{h}\left(\mathinner{\\!\left\lVert
X^{k}-\bar{X}^{k}\right\rVert}+\mathinner{\\!\left\lVert
X^{i}-\bar{X}^{i}\right\rVert}\right)$
$\displaystyle+L_{h}\epsilon^{2}\left(\frac{\delta^{3/2}\sqrt{N}}{2\delta^{3/2}\sqrt{N}-1}\right)\frac{1}{N}\sum_{m}\mathinner{\\!\left\lVert
X^{m}-\bar{X}^{m}\right\rVert}$
$\displaystyle+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty}\left(\frac{3\delta^{3/2}\sqrt{N}-1}{2\delta^{3/2}\sqrt{N}-1}\right)(\mathcal{T}_{k}+\mathcal{T}_{i}+4{\mathcal{G}}).$

Making use of (4.5) and

$\displaystyle\mathcal{G}$
$\displaystyle=\max_{j}\left(\sum_{l}T_{jl}\sum_{m}\mathinner{\\!\left\lVert
T_{lm}-\bar{T}_{lm}\right\rVert}\right)$ $\displaystyle\leq\frac{1}{(\delta
N)^{3/2}}\sum_{l}\mathcal{T}_{l},$

gives the result (4.11). ∎

##  Appendix E Proof of lemma 4.4

Throughout the proof, we will make use of the following bounds

$\displaystyle T_{ik}\mathinner{\\!\left\lVert
X^{i}-X^{k}\right\rVert}\leq\frac{g_{\epsilon}(X^{i},X^{k})}{(\delta
N)^{3/2}}\mathinner{\\!\left\lVert X^{i}-X^{k}\right\rVert}$
$\displaystyle\leq\frac{2\sqrt{d\epsilon}\exp(-1/2)}{(\delta N)^{3/2}},$ (5.1)
$\displaystyle T_{ik}\mathinner{\\!\left\lVert X^{i}-X^{k}\right\rVert}^{2}$
$\displaystyle\leq\frac{4d\epsilon\exp(-1)}{(\delta N)^{3/2}},$ (5.2)
$\displaystyle T_{ik}^{1/2}\mathinner{\\!\left\lVert
X^{i}-X^{k}\right\rVert}\leq\frac{g_{2\epsilon}(X^{i},X^{k})}{(\delta
N)^{3/4}}\mathinner{\\!\left\lVert X^{i}-X^{k}\right\rVert}$
$\displaystyle\leq\frac{2\sqrt{2\epsilon d}\exp(-1/2)}{(\delta N)^{3/4}}.$
(5.3)

We work with

$\displaystyle
K_{\epsilon}(X^{i}_{t},\rho_{t}^{N})=\frac{1}{2\epsilon}\sum_{k,l}T_{ik}T_{il}(r_{k}-r_{l})(X^{k}-X^{i}),$

since it holds that $\sum_{j}s_{ij}=0$ for all $i$. Then using Lemma 4.2, we
have that

$\displaystyle\mathinner{\\!\left\lVert
K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right\rVert}$
$\displaystyle\leq\frac{1}{2\epsilon}\sum_{k}T_{ik}^{1/2}\mathinner{\\!\left\lVert
r_{k}-r_{i}\right\rVert}T_{ik}^{1/2}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}$
$\displaystyle+\frac{1}{2\epsilon}\left(\sum_{l}T_{il}\mathinner{\\!\left\lVert
r_{i}-r_{l}\right\rVert}\right)\left(\sum_{k}T_{ik}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}\right)$
$\displaystyle\leq\frac{C_{\phi}}{2\epsilon}\sum_{k}T_{ik}^{1/2}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}T_{ik}^{1/2}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}+\frac{C_{\phi}}{2\epsilon}\left(\sum_{l}T_{il}\mathinner{\\!\left\lVert
X^{i}-X^{l}\right\rVert}\right)^{2}$ $\displaystyle\lesssim C_{\phi}d,$

since when $N$ satisfies condition 4.1, we have that
$\frac{1}{\delta^{3/2}\sqrt{N}}<2$. In order to prove the second result, we
start from

$\displaystyle\mathinner{\\!\left\lVert
K_{\epsilon}(\bar{X}^{i}_{t},\bar{\rho}_{t}^{N})-K_{\epsilon}(X_{t}^{i},\rho_{t}^{N})\right\rVert}^{2}$
$\displaystyle\leq\frac{1}{2\epsilon^{2}}\left(\sum_{k,l}\mathinner{\\!\left\lVert
T_{il}T_{ik}(r_{k}-r_{i})(X^{k}-X^{i})-\bar{T}_{il}\bar{T}_{ik}(\bar{r}_{k}-\bar{r}_{i})(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle+\frac{1}{2\epsilon^{2}}\left(\sum_{k,l}\mathinner{\\!\left\lVert
T_{il}T_{ik}(r_{i}-r_{l})(X^{k}-X^{i})-\bar{T}_{il}\bar{T}_{ik}(\bar{r}_{i}-\bar{r}_{l})(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{2\epsilon^{2}}(A_{1}+A_{2}).$

We have that

$\displaystyle A_{1}$
$\displaystyle\lesssim\left(\sum_{k,l}\mathinner{\\!\left\lVert
T_{il}-\bar{T}_{il}\right\rVert}T_{ik}\mathinner{\\!\left\lVert
r_{k}-r_{i}\right\rVert}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}\right)^{2}$
$\displaystyle+\left(\sum_{k,l}\bar{T}_{il}\mathinner{\\!\left\lVert
T_{ik}(r_{k}-r_{i})(X^{k}-X^{i})-\bar{T}_{ik}(\bar{r}_{k}-\bar{r}_{i})(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{11}+A_{12},$

and likewise

$\displaystyle A_{2}$
$\displaystyle\lesssim\left(\sum_{k,l}T_{il}\mathinner{\\!\left\lVert
r_{i}-r_{l}\right\rVert}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle+\left(\sum_{k,l}\mathinner{\\!\left\lVert\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\mathinner{\\!\left\lVert
T_{il}(r_{i}-r_{l})-\bar{T}_{il}(\bar{r}_{i}-\bar{r}_{l})\right\rVert}\right)^{2}$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{21}+A_{22}.$

Using (5.2), Lemma 4.2 (specifically, (4.10)) and Jensen’s inequality we have
that

$\displaystyle A_{11}$
$\displaystyle\lesssim\left(\sum_{l}\mathinner{\\!\left\lVert
T_{il}-\bar{T}_{il}\right\rVert}\right)^{2}\sum_{k}T_{ik}\mathinner{\\!\left\lVert
r_{k}-r_{i}\right\rVert}^{2}\mathinner{\\!\left\lVert
X^{k}-X^{i}\right\rVert}^{2}$ $\displaystyle\lesssim
C_{\gamma}^{2}d\epsilon\left(\sum_{l}\mathinner{\\!\left\lVert
T_{il}-\bar{T}_{il}\right\rVert}\right)^{2}$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}C_{\gamma}^{2}d\epsilon
B_{i}.$

Similarly, for $A_{12}$, we have that

$\displaystyle A_{12}$ $\displaystyle\leq
2\left(\sum_{k}\mathinner{\\!\left\lVert
r_{k}-r_{i}\right\rVert}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle+2\left(\sum_{k}\bar{T}_{ik}\mathinner{\\!\left\lVert\bar{X}^{i}-\bar{X}^{k}\right\rVert}\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{k}+\bar{r}_{i}\right\rVert}\right)^{2}$
$\displaystyle\leq 2C_{\gamma}^{2}\left(\sum_{k}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}$
$\displaystyle+\frac{8d\epsilon\exp(-1)}{(\delta
N)^{3/2}}\sum_{k}\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{k}+\bar{r}_{i}\right\rVert}^{2}$ $\displaystyle\lesssim
C_{\gamma}^{2}\left(\sum_{k}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}+\frac{d\epsilon}{N}\sum_{k}\mathinner{\\!\left\lVert
r_{k}-r_{i}-\bar{r}_{k}+\bar{r}_{i}\right\rVert}^{2}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}C_{\gamma}^{2}E_{i}+\frac{d\epsilon}{N}D_{i},$

and

$\displaystyle A_{21}$ $\displaystyle\leq
C_{\gamma}^{2}\left(\sum_{k}\mathinner{\\!\left\lVert
T_{ik}(X^{k}-X^{i})-\bar{T}_{ik}(\bar{X}^{k}-\bar{X}^{i})\right\rVert}\right)^{2}=C_{\gamma}^{2}E_{i},$

and using (5.1) and Jensen’s inequality

$\displaystyle A_{22}$ $\displaystyle\leq
16d\epsilon\exp(-1)\left(\sum_{l}\mathinner{\\!\left\lVert
T_{il}(r_{i}-r_{l})-\bar{T}_{il}(\bar{r}_{i}-\bar{r}_{l})\right\rVert}\right)^{2}$
$\displaystyle\lesssim
d\epsilon\left(C_{\gamma}^{2}\left(\sum_{l}\mathinner{\\!\left\lVert
T_{il}-\bar{T}_{il}\right\rVert}\right)^{2}+\frac{1}{N}\sum_{l}\mathinner{\\!\left\lVert
r_{i}-r_{l}-\bar{r}_{i}+\bar{r}_{l}\right\rVert}^{2}\right)$
$\displaystyle=d\epsilon\left(C_{\gamma}^{2}B_{i}+\frac{1}{N}D_{i}\right).$

Combining all and using Lemma 4.1 and 4.3 gives the result. ∎

##  Appendix F Proof of lemma 4.5

For notational ease, we drop the $t$ subscript throughout the proof and
consider $t<\zeta_{\delta}^{N}$. Start by analysing the summand in (4.14)

$\displaystyle\mathinner{\\!\left\lVert
K_{\epsilon}(\bar{X}^{i},\bar{\rho}^{N})-\bar{K}_{\epsilon}(\bar{X}^{i},\bar{\rho})\right\rVert}^{2}$
$\displaystyle=\frac{1}{4\epsilon^{2}}\mathinner{\\!\left\lVert\sum_{k}\bar{T}_{ik}(\bar{X}^{k}-\hat{\bar{X}}^{i})(\bar{r}^{k}-\hat{\bar{r}}^{i})-\int(\mathcal{R}_{\epsilon}(y)-{\hat{\mathcal{R}_{\epsilon}}}(\bar{X}^{i}))(y-\hat{y})p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{4\epsilon^{2}}\mathinner{\\!\left\lVert\sum_{k}T_{ik}f_{ik}-\int
f(y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert},$

where $\hat{y}\mathrel{\mathop{\mathchar 58\relax}}=\int
yp_{\epsilon}(\bar{X}^{i},y)dy$ (likewise for $\hat{\mathcal{R}}_{\epsilon}$)
and $\hat{X}^{i}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{k}T_{ik}X^{k}$ and
likewise for $\hat{r}^{i}$. For ease of notation, define
$f_{ik}\mathrel{\mathop{\mathchar
58\relax}}=(X^{k}-\hat{X}^{i})(r^{k}-\hat{r}^{i})$ and
$f\mathrel{\mathop{\mathchar
58\relax}}=(\mathcal{R}_{\epsilon}-\hat{\mathcal{R}_{\epsilon}})(y-\hat{y})$.
Now to analyse the term in brackets, we have

$\displaystyle\sum_{k}T_{ik}f_{ik}-\int f(y)p_{\epsilon}(\bar{X}^{i},y)dy$
$\displaystyle=\frac{\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})f_{ik}}{\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}-\int
f(y)p_{\epsilon}(\bar{X}^{i},y)dy$
$\displaystyle=\frac{\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})(f_{ik}-\int
f(y)p_{\epsilon}(\bar{X}^{i},y)dy)}{\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}$
$\displaystyle=\frac{\frac{1}{N}\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\int
b({\bf X},y)p_{\epsilon}(\bar{X}^{i},y)dy}{\frac{1}{N}s^{i}},$

where

$\displaystyle b({\bf X},y)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=f_{ik}-f(y)$
$\displaystyle=(X^{k}-\hat{X}^{i})(r^{k}-\hat{r}^{i})-(\mathcal{R}_{\epsilon}(y)-\hat{\mathcal{R}_{\epsilon}}(\bar{X}^{i}))(y-\hat{y}).$

From now on, we consider the expectation and $t<\zeta_{\delta}^{N}$. Using the
fact that $s^{i}\geq\delta N$ it holds that

$\displaystyle\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{\frac{1}{N}\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\int
b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dy}{\frac{1}{N}s^{i}}\right\rVert}^{2}\right]$
$\displaystyle\leq\frac{1}{\delta^{2}}\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\tilde{q}_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\int
b({\bf X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}^{2}\right]$
$\displaystyle=\frac{1}{\delta^{2}}\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\frac{g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}{(\sum_{l}g_{\epsilon}(\bar{X}^{k},\bar{X}^{l}))^{1/2}}\int
b({\bf X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}^{2}\right]$
$\displaystyle=\frac{1}{\delta^{2}N}\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\frac{g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}{(\bar{z}_{\epsilon}^{k})^{1/2}}\int
b({\bf X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}^{2}\right]$
$\displaystyle\leq\frac{2}{\delta^{2}N}\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\frac{g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})}{(g_{\epsilon}*\rho(\bar{X}^{k}))^{1/2}}\int
b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}^{2}\right]+\frac{2}{\delta^{2}N}\mathbb{E}$
$\displaystyle\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\int b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dyg_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\left(\frac{1}{(\frac{1}{N}\bar{z}_{\epsilon}^{k})^{1/2}}-\frac{1}{(g_{\epsilon}*\rho(\bar{X}^{k}))^{1/2}}\right)\right\rVert}^{2}\right]$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{2}{\delta^{2}N}(I_{1}+I_{2}),$

where $\bar{z}_{\epsilon}^{k}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{N}\sum_{l}g_{\epsilon}(\bar{X}^{k},\bar{X}^{l})$. First
note that $\int b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dyg_{\epsilon}(\bar{X}^{i},\bar{X}^{k})$ is
uniformly bounded for all $i,k$ since

$\displaystyle\mathinner{\\!\left\lVert
g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\int b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}\leq\mathinner{\\!\left\lVert
f_{ik}\right\rVert}g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})+\mathinner{\\!\left\lVert\int
f(y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert},$

and using Lemma 4.2, (4.2) and (3.4), we have that for all $i,k$,

$\displaystyle\mathinner{\\!\left\lVert
f_{ik}\right\rVert}g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\lesssim
C_{\gamma}d(\epsilon\vee\sqrt{\epsilon}).$

Secondly, under assumption (4.2) and (4.3) with $h$ uniformly bounded, we have

$\displaystyle\mathinner{\\!\left\lVert\int
f(y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}$
$\displaystyle=\mathinner{\\!\left\lVert\int\left(\mathcal{R}_{\epsilon}(y)-\int\mathcal{R}_{\epsilon}(y)p_{\epsilon}(\bar{X}^{i},y)dy\right)\left(y-\int
yp_{\epsilon}(\bar{X}^{i},y)dy\right)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}$
$\displaystyle\leq\int\mathinner{\\!\left\lVert\mathcal{R}_{\epsilon}(y)-\int\mathcal{R}_{\epsilon}(y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}\mathinner{\\!\left\lVert
y-\int
yp_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}p_{\epsilon}(\bar{X}^{i},y)dy$
$\displaystyle\lesssim(\mathinner{\\!\left\lVert\phi_{\epsilon}\right\rVert}_{\infty}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty})\delta_{v},$

where the boundedness of $\phi_{\epsilon}$ follows from the fact that the rhs
of the fixed point equation (1.11) is uniformly bounded. Combining gives

$\displaystyle\mathinner{\\!\left\lVert
g_{\epsilon}(\bar{X}^{i},\bar{X}^{k})\int b({\bf
X},y)p_{\epsilon}(\bar{X}^{i},y)dy\right\rVert}$
$\displaystyle\leq(\mathinner{\\!\left\lVert\phi_{\epsilon}\right\rVert}_{\infty}+\epsilon\mathinner{\\!\left\lVert
h\right\rVert}_{\infty})\delta_{v}+C_{\gamma}d(\epsilon\vee\sqrt{\epsilon})$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}D_{b}.$

Starting with $I_{1}$, we have

$\displaystyle I_{1}$ $\displaystyle\leq
D_{b}^{2}\mathbb{E}\left[\mathinner{\\!\left\lVert\frac{1}{N}\sum_{k}\frac{1}{(g_{\epsilon}*\rho(\bar{X}^{k}))^{1/2}}\right\rVert}^{2}\right]$
$\displaystyle\leq
D_{b}^{2}\frac{1}{N}\sum_{k}\mathbb{E}\left[\frac{1}{g_{\epsilon}*\rho(\bar{X}^{k})}\right]$
$\displaystyle\leq\frac{D_{b}^{2}}{\delta},$

since for $t<\zeta_{\delta}^{N}$, it holds that for all $k$

$\displaystyle g_{\epsilon}\ast\rho(\bar{X}^{k})\geq\delta.$ (6.1)

Then for $I_{2}$,

$\displaystyle I_{2}$ $\displaystyle\leq
D_{b}^{2}\mathbb{E}\left[\frac{1}{N}\sum_{k}\frac{\left|\bar{z}_{\epsilon}^{k}-g_{\epsilon}\ast\rho(\bar{X}^{k})\right|}{\bar{z}_{\epsilon}^{k}g_{\epsilon}\ast\rho(\bar{X}^{k})}\right]$
$\displaystyle\leq\frac{D_{b}^{2}}{\delta}\mathbb{E}\left[\frac{1}{N}\sum_{k}\frac{\left|\bar{z}_{\epsilon}^{k}-g_{\epsilon}\ast\rho(\bar{X}^{k})\right|}{g_{\epsilon}\ast\rho(\bar{X}^{k})}\right]$
$\displaystyle\leq\frac{D_{b}^{2}}{\delta}\frac{1}{N}\sum_{k}\mathbb{E}\left[\frac{1}{g_{\epsilon}\ast\rho(\bar{X}^{k})^{2}}\right]^{\frac{1}{2}}\mathbb{E}\left[\mathinner{\\!\left\lVert\bar{z}_{\epsilon}^{k}-g_{\epsilon}\ast\rho(\bar{X}^{k})\right\rVert}^{2}\right]^{\frac{1}{2}}$
$\displaystyle\leq\frac{D_{b}^{2}}{\delta}\frac{1}{N}\sum_{k}\mathbb{E}\left[\frac{1}{g_{\epsilon}\ast\rho(\bar{X}^{k})^{2}}\right]^{\frac{1}{2}}\left(\frac{1}{N}\sum_{l}\mathbb{E}\left[\mathinner{\\!\left\lVert
g_{\epsilon}(\bar{X}^{k},\bar{X}^{l})-g_{\epsilon}\ast\rho(\bar{X}^{k})\right\rVert}^{2}\right]\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\frac{D_{b}^{2}}{\delta}\frac{1}{N}\sum_{k}\mathbb{E}\left[\frac{1}{g_{\epsilon}\ast\rho(\bar{X}^{k})^{2}}\right]^{\frac{1}{2}},$

where the last inequality arises from the fact that for all $l,k$, it holds
that

$\displaystyle\mathbb{E}\left[\mathinner{\\!\left\lVert
g_{\epsilon}(\bar{X}^{k},\bar{X}^{l})-g_{\epsilon}\ast\rho(\bar{X}^{k})\right\rVert}^{2}\right]$
$\displaystyle\leq 2\mathbb{E}\left[\mathinner{\\!\left\lVert
g_{\epsilon}(\bar{X}^{k},\bar{X}^{l})\right\rVert}^{2}\right]+2\mathbb{E}\left[\mathinner{\\!\left\lVert
g_{\epsilon}\ast\rho(\bar{X}^{k})\right\rVert}^{2}\right]$ $\displaystyle\leq
4.$

To analyse the remaining expectation, we have that using (6.1) for every $k$,

$\displaystyle\mathbb{E}\left[\frac{1}{g_{\epsilon}\ast\rho(\bar{X}^{k})^{2}}\right]\leq\frac{1}{\delta^{2}}.$

Combining leads to

$\displaystyle I_{2}\leq\frac{D_{b}^{2}}{\delta^{2}}.$

the result then follows easily. ∎

##  Appendix G Some useful theorems

######  Lemma G.1.

Generalised Brascamp-Lieb inequality, theorem 1.1 in [CCeL13] Let
$g,h\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}$
be square integrable locally Lipschitz functions and suppose $d\mu=\exp(-V)dx$
is a strictly log-concave measure. It then holds that

$\displaystyle\mathinner{\\!\left\lVert Cov_{\mu}(g,h)\right\rVert}$
$\displaystyle\leq\mathinner{\\!\left\lVert Hess(V)^{-1/2}\nabla
g\right\rVert}_{2}\mathinner{\\!\left\lVert Hess(V)^{-1/2}\nabla
h\right\rVert}$ $\displaystyle=\left(\int\nabla g^{T}Hess(V)^{-1}\nabla
gd\mu\right)^{1/2}\left(\int\nabla h^{T}Hess(V)^{-1}\nabla
hd\mu\right)^{1/2},$

where the norms on the rhs refer to the euclidean norm (although the result
holds for more general $L^{p}$ norms.)

Suppose furthermore that $V$ is uniformly convex, i.e.
$\inf_{x}y^{T}Hess(V)(x)y\geq\kappa|y|^{2}$ for all $y\in\mathbb{R}^{d}$, then

$\displaystyle\mathinner{\\!\left\lVert Cov_{\mu}(g,h)\right\rVert}$
$\displaystyle\leq\frac{1}{\kappa}\left(\int\mathinner{\\!\left\lVert\nabla
g\right\rVert}^{2}d\mu\right)^{1/2}\left(\int\mathinner{\\!\left\lVert\nabla
h\right\rVert}^{2}d\mu\right)^{1/2}.$

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