# On nonparametric estimation of the interaction function in particle system
models

Denis Belomestny1 1Faculty of Mathematics
Duisburg-Essen University
Thea-Leymann-Str. 9
D-45127 Essen
Germany<EMAIL_ADDRESS>,  Mark Podolskij2 2 Department of
Mathematics, University of Luxembourg
Luxembourg<EMAIL_ADDRESS>and  Shi-Yuan Zhou3 3 Department of
Mathematics, University of Luxembourg
Luxembourg<EMAIL_ADDRESS>
###### Abstract.

This paper delves into a nonparametric estimation approach for the interaction
function within diffusion-type particle system models. We introduce two
estimation methods based upon an empirical risk minimization. Our study
encompasses an analysis of the stochastic and approximation errors associated
with both procedures, along with an examination of certain minimax lower
bounds. In particular, we show that there is a natural metric under which the
corresponding minimax estimation error of the interaction function converges
to zero with parametric rate. This result is rather suprising given complexity
of the underlying estimation problem and rather large classes of interaction
functions for which the above parametric rate holds.

Keywords: drift estimation, McKean-Vlasov diffusions, non-linear SDEs,
nonparametric statistics, particle systems.

AMS 2010 subject classification: 62G20,62M05, 60G07, 60H10.

##  1\. Introduction

This article is dedicated to the study of $N$-dimensional interacting particle
systems described by the equation

(1) $dX_{t}^{i,N}=(\varphi\star\mu_{t}^{N})(X_{t}^{i,N})\,dt+\sigma
dW_{t}^{i},\quad i=1,\ldots,N,$

where $t\in[0,T],$ $\varphi\colon\mathbb{R}\to\mathbb{R}$ is the interaction
potential and $(W^{i})_{i=1}^{N}$ are independent one-dimensional Brownian
motions. Here, $\mu_{t}^{N}$ stands for the empirical measure of the particle
system at time $t$, given by

(2) $\mu_{t}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i,N}},$

and $\varphi\star\mu(x):=\int_{\mathbb{R}}\varphi(x-y)\mu(dy)$. We make the
assumption that we observe $N$ paths $(X_{t}^{i,N},t\in[0,T],i=1,\ldots,N)$
and aim to estimate the unknown interaction function $\varphi$ as $N\to\infty$
with $T>0$ being fixed. It is noteworthy that, given complete path
observations, the diffusion coefficient $\sigma$ can be considered known.

Interacting particle systems of the type Eq. 1 play an important role in
probability theory and applications. Initially introduced by McKean in
pioneering work [34], these systems served as models for plasma dynamics. In
more recent years, diffusion-type interacting particle systems have found
wide-ranging applications in finance [9, 20, 21, 26, 27], social science [12],
neuroscience [4], and population dynamics [35], among others. From a
probabilistic perspective, the particle system delineated in Eq. 1 is
inherently linked to its corresponding mean-field limiting equation—a one-
dimensional McKean-Vlasov stochastic differential equation (SDE),

(3) $\left\\{\begin{array}[]{ll}dX_{t}=(\varphi\star\mu_{t})(X_{t})\,dt+\sigma
dB_{t},\\\ \textnormal{Law}(X_{t})=\mu_{t},\end{array}\right.$

where $B$ is a $1$-dimensional Brownian motion. Under suitable assumptions on
the function $\varphi$, a well-known phenomenon known as the propagation of
chaos emerges. This phenomenon indicates that, as $N\rightarrow\infty$, the
empirical probability measure $\mu_{t}^{N}$ weakly converges to $\mu_{t}$ for
any $t>0$, and the McKean-Vlasov stochastic differential equation (SDE) Eq. 3
characterizes the asymptotic trajectory of an individual particle. Classical
treatments of the propagation of chaos and McKean-Vlasov SDEs can be found in
the monographs [31, 40]. Furthermore, the measure $\mu_{t}$ is absolutely
continuous with respect to the Lebesgue measure for all $t>0$, assuming
certain regularity conditions on $\varphi$. Moreover, the density of
$\mu_{t}$, denoted again by $\mu_{t}$ (with a slight abuse of notation),
satisfies the granular media equation:

(4)
$\displaystyle\partial_{t}\mu_{t}=\partial_{x}\\{(\sigma^{2}/2)\partial_{x}\mu_{t}+(\varphi\star\mu_{t})\mu_{t}\\}.$

The realm of statistical inference for interacting particle systems remains a
less explored domain. While a considerable number of recent articles delve
into parametric estimation for particle systems and McKean-Vlasov stochastic
differential equations under diverse model and sampling assumptions (see, for
instance, [1, 7, 13, 16, 18, 24, 25, 30, 33, 38, 44]), non- and semiparametric
estimation of the interaction potential $\varphi$ presents a notably more
intricate challenge. This complexity arises from the convolution with measures
$\mu_{t}^{N}$ or $\mu_{t}$ and has only been the subject of investigation in
three recent papers. Specifically, the articles [2, 5] explore the
semiparametric estimation of $\varphi$ given observations
$X_{T}^{i,N},i=1,\ldots,N,$ with $N,T\to\infty$. The methodology in [2, 5]
heavily relies on the existence of the invariant density, necessitating the
asymptotic regime $T\to\infty,$ and employs the deconvolution method. Another
relevant work, [17], focuses on the same observation scheme as the current
paper, presenting a nonparametric approach for the drift function
$\varphi\star\mu_{t}$ but demonstrating consistency only for an estimator of
the interaction function $\varphi$.

This paper aims to investigate nonparametric estimation of the function
$\varphi$ using the sampling scheme $X_{t}^{i,N},t\in[0,T],i=1,\ldots,N,$ with
$N\to\infty$ and $T$ held fixed. We propose two interconnected methods, both
based on empirical risk minimization over sieves $(S_{m})_{m\geq 1}$,
employing an unconventional underlying loss function $\|\cdot\|_{\star}$ that
depends on the unknown measure $(\mu_{t})_{t\in[0,T]}$. Our first approach
addresses the scenario where the functional spaces $(S_{m})_{m\geq 1}$ are
compact with respect to $\|\cdot\|_{\infty}$. In this case, the analysis of
the proposed estimator’s rate heavily relies on empirical processes theory for
(degenerate) U-statistics. The second approach investigates a more traditional
setting, where $(S_{m})_{m\geq 1}$ are finite-dimensional vector spaces. Here,
we require a further truncation of the empirical risk minimizer, akin to the
approach introduced in [15] for classical stochastic differential equations,
where the coefficients are independent of the laws $(\mu_{t})_{t\in[0,T]}$. We
derive convergence rates with respect to the loss function
$\|\cdot\|_{\star}$, involving an examination of stochastic and approximation
errors. We will demonstrate that the resulting convergence rate is often
$\sqrt{\log(N)/N}$ in the first scenario, which essentially matches the
minimax lower bound. However, for the second method, the convergence rate
decreases due to the truncation procedure. Additionally, we will establish
that the minimax lower bound for the traditional $L^{2}$-loss is logarithmic,
consistent with the findings of [5].

The paper is structured as follows. Section 2 concentrates on a risk
minimization approach over compact functional classes. This section elucidates
the core concepts, establishes convergence rates in terms of metric entropy,
and analyses the approximation error associated with the estimation method. In
Section 3, we introduce an alternative approach that involves risk
minimization over vector spaces coupled with an additional dimension
truncation. Uniform convergence rates for the resulting estimator are
presented. Section 4 delves into an exploration of minimax lower bounds.
Specifically, we demonstrate that the estimator developed in Section 2 attains
essentially optimal rates with respect to the $\|\cdot\|_{\star}$-norm.
Additionally, we establish a logarithmic lower bound when considering the
traditional $L^{2}$-loss. For the detailed proofs of the main results, we
refer to Section 5.

#### Notation

All random variables and stochastic processes are defined on a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$. Throughout, positive constants are represented by $C$ (or
$C_{p}$ if dependent on an external parameter $p$), though their values may
vary across lines. We use the notation $C^{k}(\mathbb{R})$ (resp.
$C_{c}^{2}(\mathbb{R})$) to denote the space of $k$ times continuously
differentiable functions (resp. the space of $k$ times continuously
differentiable functions with compact support). The sup-norm of a function
$f:\mathbb{R}\to\mathbb{R}$ is denoted as $\|f\|_{\infty}$ or
$\|f\|_{Q}:=\sup_{x\in Q}|f(x)|$. When referring to a probability measure or
density $\mu$, $\widehat{\mu}$ denotes its Fourier transform:

$\widehat{\mu}(z)=\int_{\mathbb{R}}\exp(izx)\mu(x)\,dx.$

Given a matrix $A\in\mathbb{R}^{k\times k}$, we denote by $\|A\|_{\text{op}}$
the operator norm of $A$; we write $A\succcurlyeq B$ if $A-B$ is a positive
semidefinite matrix. For two sequences $(a_{n})_{n\in\mathbb{N}}$ and
$(b_{n})_{n\in\mathbb{N}}$ we write $a_{n}\lesssim b_{n}$ if there exists a
constant $C>0$ with $a_{n}\leq Cb_{n}$ for all $n\in\mathbb{N}$. The Orlicz
norm of a real valued random variable $\eta$ with respect to a nondecreasing,
convex function $\psi$ on $\mathbb{R}$ with $\psi(0)=0$ is defined by

$\|\eta\|_{\psi}:=\inf\left\\{t>0:\mathbb{E}\psi\bigl{(}|\eta|/t\bigr{)}\leq
2\right\\}.$

When $\psi_{p}(x):=x^{p}$ with $p\geq 1$ the corresponding Orlicz norm is (up
to a constant) the $L^{p}$-norm $\|\eta\|_{p}=(\mathbb{E}[|\eta|^{p}])^{1/p}.$
We say that $\eta$ is _sub-Gaussian_ if $\|\eta\|_{\psi_{e,2}}<\infty$ with
$\psi_{e,2}:=\exp(x^{2})-1.$ In particular, this implies that for some
constants $C,c>0$,

$\mathbb{P}(|\eta|\geq t)\leq
2\exp\left(-\frac{ct^{2}}{\|\eta\|_{\psi_{e,2}}^{2}}\right)\text{\ \ and \ \
}\mathbb{E}[|\eta|^{p}]^{1/p}\leq C\sqrt{p}\|\eta\|_{\psi_{e,2}},\quad p\geq
1.$

Consider a real valued random process $(X_{t})_{t\in\mathcal{T}}$ on a metric
space $(\mathcal{T},\mathsf{d})$. We say that the process has _sub-Gaussian
increments_ if there exists $C>0$ such that

$\|X_{t}-X_{s}\|_{\psi_{e,2}}\leq C\mathsf{d}(t,s),\quad\forall
t,s\in\mathcal{T}.$

Let $(\mathsf{Y},\rho)$ be a metric space and $\mathsf{X\subseteq Y}$. For
$\varepsilon>0$, we denote by $\mathcal{N}(\mathsf{X},\rho,\varepsilon)$ the
covering number of the set $\mathsf{X}$ with respect to the metric $\rho$,
that is, the smallest cardinality of a set (or net) of $\varepsilon$-balls in
the metric $\rho$ that covers $\mathsf{X}$. Then
$\log\mathcal{N}(\mathsf{X},\rho,\varepsilon)$ is called the metric entropy of
$\mathsf{X}$ and

$\mathsf{DI}(\mathsf{X},\rho,\psi):=\int_{0}^{\operatorname{diam}(\mathsf{X})}\psi^{-1}\bigl{(}\mathcal{N}(\mathsf{X},\rho,u)\bigr{)}\,du$

with
$\operatorname{diam}(\mathsf{X}):=\max_{x,x^{\prime}\in\mathsf{X}}\rho(x,x^{\prime}),$
is called the Dudley integral with respect to $\psi.$ For example, if
$|\mathsf{X}|<\infty$ and $\rho(x,x^{\prime})=1_{\\{x\neq x^{\prime}\\}}$ we
get $\mathsf{DI}(\mathsf{X},\rho,\psi_{e,2})=\sqrt{\log|\mathsf{X}|}.$

##  2\. Risk minimization over compact functional classes

We start by introducing assumptions on the underlying interacting particle
system that insure the propagation of chaos phenomenon. We assume that the
initial law of the model is given by
$\textnormal{Law}(X_{0}^{1,N},\ldots,X_{0}^{N,N})=\mu_{0}^{\otimes N}$, where
$\mu_{0}^{\otimes N}$ denotes the $N$-fold product measure of some probability
measure $\mu_{0}$ with $\int x^{k}\mu_{0}(dx)<\infty$ for all
$k\in\mathbb{N}$. Furthermore, we assume the following condition.

######  Assumption 1.

The interaction function $\varphi:\mathbb{R}\to\mathbb{R}$ is globally
Lipschitz continuous and bounded, that is,

$\displaystyle|\varphi(x)-\varphi(y)|\leq
L_{\varphi}|x-y|,\quad|\varphi(x)|\leq K_{\varphi},\quad x,y\in\mathbb{R}$

for some finite $L_{\varphi},K_{\varphi}>0.$

This condition guarantees the validity of propagation of chaos in the sense
that, for all $t\in[0,T]$, $\mu_{t}^{N}$ converges weakly to $\mu_{t}$ (see
e.g. [11, Theorem 3.1]).

###  2.1. Construction of the estimator and main results

We initiate our exposition by outlining the fundamental principles of our
estimation methodology. To begin, we consider a sequence of spaces
$(S_{m})_{m\geq 1}$ (sieves). The crucial idea of our approach lies in the
following minimization strategy:

(5) $\min_{f\in
S_{N}}\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}\left(f\star\mu_{t}^{N}(X_{t}^{i,N})-\varphi\star\mu_{t}^{N}(X_{t}^{i,N})\right)^{2}dt.$

However, the above risk function cannot be directly computed from the data
since the interaction function $\varphi$ is unknown. We derive an empirical
(noisy) version of the minimization problem by omitting the irrelevant term
$\varphi\star\mu_{t}^{N}(X_{t}^{i,N})^{2}$ in the integrand and minimizing the
resulting quantity:

$\displaystyle\gamma_{N}(f):=\frac{1}{NT}\sum_{i=1}^{N}\left(\int_{0}^{T}f\star\mu_{t}^{N}(X_{t}^{i,N})^{2}\,dt-2\int_{0}^{T}f\star\mu_{t}^{N}(X_{t}^{i,N})\,dX_{t}^{i,N}\right)$

over $S_{N}$. For further analysis, we introduce the following bilinear forms:

$\displaystyle\langle f,g\rangle_{N}$
$\displaystyle:=\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}(f\star\mu_{t}^{N})(X_{t}^{i,N})(g\star\mu_{t}^{N})(X_{t}^{i,N})\,dt,$
$\displaystyle\langle f,g\rangle_{\star}$
$\displaystyle:=\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}(f\star\mu_{t})(x)(g\star\mu_{t})(x)\mu_{t}(x)\,dx\,dt.$

We set $\|f\|_{N}^{2}:=\langle f,f\rangle_{N}$ and $\|f\|_{\star}^{2}:=\langle
f,f\rangle_{\star}$. With these notations at hand, we deduce the identity

(6)
$\|f-\varphi\|_{N}^{2}=\gamma_{N}(f)+\gamma^{\prime}_{N}(f)+\|\varphi\|_{N}^{2}$

where $\gamma^{\prime}_{N}(f)$ is the martingale term defined via

$\gamma^{\prime}_{N}(f):=\frac{2\sigma}{NT}\sum_{i=1}^{N}\int_{0}^{T}(f\star\mu_{t}^{N})(X_{t}^{i,N})\,dW_{t}^{i}.$

Finally, our estimator $\varphi_{N}$ is defined as follows:

(7) $\varphi_{N}:=\text{argmin}_{f\in
S_{N}}\gamma_{N}(f),\qquad\varphi_{\star}:=\text{argmin}_{f\in
S_{N}}\|f-\varphi\|_{\star}^{2}.$

The identity (6) means that by minimizing $\gamma_{N}(f)$ we minimize (up to a
martingale term $\gamma^{\prime}_{N}(f)$) $\|f-\varphi\|_{N}$ which is close
to $\|f-\varphi\|_{\star}$ for $N$ large enough.

Before we proceed with the asymptotic analysis, we give some remarks about a
rather unconventional loss function $\|\cdot\|_{\star}$ and its relation to
the classical $L^{2}(\mathbb{R})$-norm.

######  Remark 2.1.

The upcoming analysis will reveal that the risk $\|\cdot\|_{\star}$ is the
intrinsic norm for evaluating error bounds. Unlike conventional literature on
empirical risk minimization, it is crucial to note that this norm is elusive
since the laws $\mu_{t}$ remain unobserved. This presents one of the principal
mathematical challenges in our statistical analysis. ∎

######  Remark 2.2.

The norm $\|\cdot\|_{\star}$ can be bounded from above by the
$L^{2}(\mathbb{R})$-norm. Indeed, if we assume that the densities
$(\mu_{t})_{t\in[0,T]}$ satisfy
$\int_{0}^{T}\|\mu_{t}\|_{\infty}\,dt<{\infty}$, we conclude by the Young’s
convolution inequality that

$\|f\|_{\star}^{2}\leq\frac{1}{T}\int_{0}^{T}\|\mu_{t}\|_{\infty}\|f\star\mu_{t}\|^{2}_{L^{2}(\mathbb{R})}\,dt\leq\|f\|^{2}_{L^{2}(\mathbb{R})}\frac{1}{T}\int_{0}^{T}\|\mu_{t}\|_{\infty}\,dt.$

Nonetheless, these two norms are far from being equivalent. To illustrate this
point, we assume, for simplicity, that $\mu_{t}=\mu$ represents the density of
the standard normal distribution for all $t$. Now, consider the function
$f(x)=1_{[2M,3M]}(x)$ for some $M>0$. In this scenario, it is evident that
$\|f\|_{L^{2}(\mathbb{R})}^{2}=M$. On the other hand, there exist constants
$c_{1},c_{2}>0$ such that

$\displaystyle\|f\|_{\star}^{2}$
$\displaystyle=\int_{[-M,M]}(f\star\mu)^{2}(x)\mu(x)\,dx+\int_{[-M,M]^{c}}(f\star\mu)^{2}(x)\mu(x)\,dx$
$\displaystyle\leq c_{1}\exp(-c_{2}M^{2}).$

In particular, $\|f\|_{L^{2}(\mathbb{R})}\to\infty$ while $\|f\|_{\star}\to 0$
as $M\to\infty$ at an exponential rate. ∎

To formulate our main statements, we assume that the spaces $(S_{m})_{m\geq
1}$ are compact with respect to $\|\cdot\|_{\infty}$ with $\|f\|_{\infty}\leq
K_{\varphi}$ for $f\in S_{m}$. In certain cases, we employ the notation
$S_{m}(K_{\varphi})$ to explicitly express the dependency of spaces on the
parameter $K_{\varphi}$. We assume that the following property holds:

$f,g\in S_{m}(K_{\varphi})\implies f\pm g\in S_{m}(2K_{\varphi}).$

Subsequently, we establish the bounds for the error term
$\|\varphi_{N}-\varphi\|_{\star}.$

######  Theorem 2.3.

It holds under Assumption 1,

$\Bigl{\\{}\mathbb{E}\bigl{[}\|\varphi_{N}-\varphi\|^{q}_{\star}\bigr{]}\Bigr{\\}}^{1/q}\leq\|\varphi_{\star}-\varphi\|_{\star}+C\left(\frac{K_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{e,2})}{\sqrt{N}}\right.\\\
\left.+\frac{p^{2}K_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{p/2})}{N}+\frac{p^{3}K_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{p/3})}{N^{3/2}}\right)$

for any $p>2,$ $2\leq q\leq p,$ where $C>0$ is an absolute constant and
$\varphi^{\star}$ stands for the best approximation of $\varphi$ as defined in
(7).

Let us outline the key concepts behind the proof of Theorem 2.3. We commence
with the inequality:

(8)
$\|\varphi_{N}-\varphi\|_{\star}^{2}\leq\|\varphi_{\star}-\varphi\|_{\star}^{2}+2\left(\sup_{f\in
S_{N}}\left|\|f-\varphi\|_{N}^{2}-\|f-\varphi\|_{\star}^{2}\right|+\sup_{f\in
S_{N}}|\gamma^{\prime}_{N}(f)|\right).$

This inequality decomposes the estimation error into an approximation error
$\|\varphi_{\star}-\varphi\|_{\star}^{2}$ and stochastic errors related to the
last two terms. To handle the stochastic errors, we initially consider i.i.d.
observations from the McKean-Vlasov SDE (3). Remarkably, the term
$\|f-\varphi\|_{N}^{2}-\|f-\varphi\|_{\star}^{2}$ turns out to be a linear
combination of (degenerate) U-statistics, while $\gamma^{\prime}_{N}(f)$
represents a martingale, as noted earlier. Concentration inequalities are
applied to both terms, leading to the moment bound in Theorem 2.3 via the
metric entropy. In the final step of the proof, we employ a change of measure
device initially proposed in [17] to transfer the statement from i.i.d.
observations of the McKean-Vlasov SDE (3) to observations of the original
particle system (1). In case we have a bound on the covering number of the
space $S_{N}$, we immediately obtain the following corollary.

######  Corollary 2.4.

Suppose that
$\mathcal{N}\bigl{(}S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\varepsilon\bigr{)}\lesssim\varepsilon^{-D_{N}}$
for all $N$ and an increasing sequence of positive numbers $D_{N}$ satisfying
$D^{2}_{N}N^{-1/2}\to 0$ as $N\to\infty.$ Then it holds

$\displaystyle\Bigl{\\{}\mathbb{E}\bigl{[}\|\varphi_{N}-\varphi\|^{q}_{\star}\bigr{]}\Bigr{\\}}^{1/q}\leq\|\varphi_{\star}-\varphi\|_{\star}+CK_{\varphi}\,\sqrt{\frac{D_{N}}{N}}.$

Our focus now shifts to examining the approximation error
$\|\varphi_{\star}-\varphi\|_{\star}$, which will be addressed in the
subsequent analysis.

###  2.2. Approximation error

In this segment, we delve into an examination of the approximation error
outlined in Theorem 2.3. The strong contractivity inherent in the norm
$\|\cdot\|_{\star}$ introduces a surprising characteristic wherein the error
$\|\varphi_{\star}-\varphi\|_{\star}$ often exhibits exponential decay in the
dimensionality of the space $S_{N}$. In the following, we adopt the subsequent
assumption regarding the initial distribution:

######  Assumption 2.

Assume that the initial distribution is normal, that is,

$\mu_{0}(x)=\frac{1}{\sqrt{2\pi\zeta^{2}}}\exp(-x^{2}/(2\zeta^{2}))$

for some $\zeta>0.$

We also consider a condition on the interaction function $\varphi$:

######  Assumption 3.

There is a monotone increasing sequence $(\alpha_{n})$ with

$\alpha_{n}\to\infty,\quad\alpha_{n+1}/\alpha_{n}\to 1,\quad n\to\infty$

such that the following representation holds

$\displaystyle(\mu_{t}\star\varphi)(x)=\frac{1}{\sqrt{\alpha_{n}}}\sum_{k=0}^{\infty}c_{k}\exp(i\pi
kx/\alpha_{n})\widehat{\mu}_{t}(-k/\alpha_{n}),\quad
x\in[-\alpha_{n},\alpha_{n}]$

with coefficients $c_{k}=c_{k}(\alpha_{n})$ satisfying
$\sum_{k=1}^{\infty}|c_{k}|\lesssim K_{\varphi}\sqrt{\alpha_{n}}.$

######  Remark 2.5.

Let us examine Assumption 3. This assumption holds for any function $\varphi$,
which admits a Fourier series representation of the form:

(9)
$\displaystyle\varphi(x)=\frac{1}{\sqrt{\alpha}}\sum_{k=0}^{\infty}a_{k}\exp(i\pi
kx/\alpha),\quad x\in\mathbb{R},$

with absolutely summable coefficients $(a_{k})_{k\geq 0}$ satisfying
$\sum_{k}|a_{k}|\leq\sqrt{\alpha}K_{\varphi}$ and some $\alpha>0.$ This type
of periodic potentials appear in McKean-Vlasov equations arising in physics,
see e.g. the Kuramoto-Shinomoto-Sakaguchi model in [23]. Indeed, for such
functions we readily deduce the identity

$(\mu_{t}\star\varphi)(x)=\frac{1}{\sqrt{\alpha}}\sum_{k=0}^{\infty}a_{k}\exp(i\pi
kx/\alpha)\widehat{\mu}_{t}(-k/\alpha).$

Taking, for instance, $\alpha_{n}=\alpha n$ and comparing the coefficients, we
obtain the statement of Assumption 3 with $c_{kn}=a_{k}\sqrt{n}$ and $c_{l}=0$
for $l\not\in\\{kn:~{}k\geq 0\\}$. ∎

Next, for any $A>1,$ $D\in\mathbb{N},$ we consider the functional space

$S(A,D):=\left\\{x\to\frac{1}{\sqrt{A}}\sum_{k=0}^{D}c_{k}\exp(i\pi
kx/A):\quad\sum_{k=1}^{D}|c_{k}|\leq\sqrt{A}K_{\varphi}\right\\}.$

The main result of this subsection is the following theorem.

######  Theorem 2.6.

Set with arbitrary small $\delta>0,$

$A_{N}:=\sqrt{2(\zeta^{2}+T)\log(N)},\qquad
D_{N}:=\sqrt{(2+\delta)\frac{\zeta^{2}+T}{\zeta^{2}}}\log(N)$

and define $n_{N}:=\min\\{n:\alpha_{n}\geq A_{N}\\},$ $N\in\mathbb{N}.$ Then
under Assumptions 2 and 3, we obtain

$\displaystyle\inf_{g\in
S_{N}=S(\alpha_{n_{N}},D_{N})}\|g-\varphi\|_{\star}^{2}\lesssim\frac{\exp(K_{\varphi}^{2}T)}{N}.$

As a consequence, we deduce that

$\displaystyle\Bigl{\\{}\mathbb{E}\bigl{[}\|\varphi_{N}-\varphi\|^{2}_{\star}\bigr{]}\Bigr{\\}}^{1/2}\lesssim\sqrt{\frac{\exp(K_{\varphi}^{2}T)}{N}}+K_{\varphi}\,\sqrt{\frac{\log(N)}{N}}.$

Here $\lesssim$ stands for inequality up to an absolute constant.

As a consequence of Theorem 2.6, the resulting convergence rate is established
as $\sqrt{\log(N)/N}$. We will demonstrate in Section 4 that this rate is
essentially optimal. However, it is important to note, as highlighted in
Remark 2.2, that this does not necessarily imply a polynomial convergence rate
with respect to the classical $\|\cdot\|_{L^{2}(\mathbb{R})}$-norm. In fact,
we will establish in Section 4 that the minimax lower bound for the
$\|\cdot\|_{L^{2}(\mathbb{R})}$-norm is logarithmic, consistent with the
findings of [5].

######  Remark 2.7.

Theorem 2.6 suggests that the approximation error decays exponentially when
dealing with functions of the form described in Eq. 9. However, this
phenomenon is not exclusive to the Fourier basis. Remarkably, a similar result
can be observed for the polynomial basis. The forthcoming explanation sheds
light on the fast rates of approximation in the $\|\cdot\|_{\star}$-norm.
Suppose that $\varphi\in L^{1}(\mathbb{R})$. Then we have by the Parseval
identity

$(\varphi\star\mu_{t})(x+iy)=\frac{1}{2\pi}\int_{\mathbb{R}}\exp(iux-
uy)\widehat{\varphi}(u)\widehat{\mu}_{t}(-u)\,du.$

Under Assumption 2, $|\widehat{\mu}_{t}(u)|\leq\exp(-u^{2}\zeta^{2}/2)$ (see
Eq. 35) and hence

$\displaystyle|(\varphi\star\mu_{t})(x+iy)|$
$\displaystyle\leq\frac{\|\varphi\|_{L^{1}}}{2\pi}\int\exp(-uy)|\widehat{\mu}_{t}(u)|\,du$
$\displaystyle\leq\sqrt{\frac{1}{2\pi\zeta^{2}}}\|\varphi\|_{L^{1}}\exp\left(y^{2}/(2\zeta^{2})\right).$

Thus

$\max_{|z|=r}|(\varphi\star\mu_{t})(z)|\leq\sqrt{\frac{1}{2\pi\zeta^{2}}}\|\varphi\|_{L^{1}}\exp\left(r^{2}/(2\zeta^{2})\right),\quad
r>0.$

As a result, the function $\varphi\star\mu_{t}$ is entire of order $2$ and
finite type. Furthermore, we deduce the power series expansion

$(\varphi\star\mu_{t})(z)=\sum_{n=0}^{\infty}c_{n}z^{n}$

with $c_{n}\leq\left(e/\zeta^{2}n\right)^{n/2}.$ This implies the inequality

$\displaystyle\sup_{x\in[-A,A]}\left|(\varphi\star\mu_{t})(x)-\sum_{n=0}^{D}c_{n}x^{n}\right|$
$\displaystyle\leq\varepsilon_{D,A}=\sum_{n=D+1}^{\infty}\left(\frac{eA^{2}}{\zeta^{2}n}\right)^{n/2}$

Note that

$\varepsilon_{D,A}\leq 2e^{-(1+D)/2}$

provided $(1+D)^{-1}A^{2}\leq\zeta^{2}/e^{2}.$ Let $T_{K}[f]=1_{\\{|f(x)|\leq
K\\}}f(x)+K\mathrm{sign}(f(x))$ be a truncation operator at level $K$. Then it
holds that

$\inf_{\psi\in\mathrm{span}\\{z^{n},n\leq
D\\}}\frac{1}{T}\int_{0}^{T}\int\left((\mu_{t}\star\varphi)(x)-T_{K_{\varphi}}[\psi\star\mu_{t}](x)\right)^{2}\mu_{t}(x)\,dx\,dt\leq\varepsilon_{D,A}^{2}+R_{T}(A)$

with

$R_{T}(A)=\frac{4K_{\varphi}^{2}}{T}\int_{0}^{T}\int_{|x|>A}\mu_{t}(x)\,dx\,dt.$

Here we used the fact that for any $\psi\in\mathrm{span}\\{z^{n}:~{}n\leq
D\\}$ it holds $\psi\star\mu_{t}\in\mathrm{span}\\{z^{n}:~{}n\leq D\\}.$
Similarly to the case of Fourier basis (see proof of Theorem 2.6), we get

$R_{T}(A)\leq
8K_{\varphi}^{2}\exp\left(K_{\varphi}^{2}T/2\right)\exp\left(-\frac{A^{2}}{2(\zeta^{2}+T)}\right).$

If we chose $A_{N}=\sqrt{2(\zeta^{2}+T)\log(N)}$ and
$D_{N}=\frac{2e^{2}(\zeta^{2}+T)}{\zeta^{2}}\log(N)$, then

$\inf_{\varphi\in\mathrm{span}\\{z^{n}:~{}n\leq
D_{N}\\}}\frac{1}{T}\int_{0}^{T}\int\left((\mu_{t}\star\varphi)(x)-T_{K_{\varphi}}[\psi\star\mu_{t}](x)\right)^{2}\mu_{t}(x)\,dx\,dt\lesssim\frac{\exp\left(K_{\varphi}^{2}T/2\right)}{N}.$

In other words, we obtain an exponential decay of the approximation error for
the polynomial basis as well. ∎

##  3\. Risk minimization over vector spaces

While the empirical minimization approach presented in the preceding section
serves as a valuable estimation method, it is not without its limitations.
Firstly, the construction of the estimator $\varphi_{N}$ in (7) inherently
assumes knowledge of the constant $K_{\varphi}$. Secondly, a significant
drawback lies in the necessity of the compactness property of functional
spaces $S_{m}$ with respect to the sup-norm, a condition that is notably
stringent.

###  3.1. Setting and construction of the estimator

This section focuses on a more traditional setting of risk minimization within
vector spaces. We assume that the finite-dimensional spaces $S_{N}$ are
equipped with an inner product $\langle\cdot,\cdot\rangle$, and
$(e_{1},\ldots,e_{D_{N}})$ represents an orthonormal basis of $S_{N}$
satisfying $\|e_{j}\|_{\infty}<\infty$ for all $j=1,\ldots,D_{N}$.

Due to the vector space structure, the estimator $\varphi_{N}$ introduced at
(7) can be computed explicitly as

(10) $\varphi_{N}=\sum_{j=1}^{D_{N}}\left(\theta_{N}\right)_{j}e_{j},$

where $\theta_{N}=\Psi_{N}^{-1}Z_{N}$ and the random vectors $\Psi_{N},\
Z_{N}$, taking values in $\mathbb{R}^{D_{N}\times D_{N}}$ and
$\mathbb{R}^{D_{N}}$ respectively, are given by

$\displaystyle(\Psi_{N})_{jk}$ $\displaystyle=\langle e_{j},e_{k}\rangle_{N},$
(11) $\displaystyle(Z_{N})_{j}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}e_{j}\star\mu_{t}^{N}(X_{t}^{i,N})\,dX_{t}^{i,N}.$

There is a theoretical counterpart to the empirical matrix $\Psi_{N}$. Indeed,
we have that $\mathbb{E}{\Psi_{N}}\approx\Psi$ where the matrix $\Psi$ is
given by the formula

(12) $(\Psi)_{jk}=\langle e_{j},e_{k}\rangle_{\star}.$

Obviously, both matrices are positive semidefinite by construction. In the
next step, we will regularise the estimator $\varphi_{N}$. For a given
sequence $D_{N}$, the operator norm $\|\Psi_{N}^{-1}\|_{\text{op}}$ may become
too large compared to the sample size $N$ and we would like to avoid such
situations. For this purpose, we restrict the growth of
$\|\Psi^{-1}\|_{\text{op}}.$

######  Assumption 4.

Let $\eta\geq 5$ be a given number. We assume that $\Psi_{N}$ is invertible
almost surely and the sequence $D_{N}$ is such that the following growth
condition is satisfied:

$L_{N}^{2}\|\Psi^{-1}\|_{\text{op}}^{2}\leq\frac{c_{\eta,T}NT}{4\log(NT)},$

where $L_{N}:=\sum_{j=1}^{D_{N}}\|e_{j}\|^{2}_{\infty}$ and
$c_{\eta,T}:=(72\eta T)^{-1}$.

We now consider the regularised version of the initial estimator $\varphi_{N}$
introduced at Eq. 7:

(13)
$\widehat{\varphi}_{N}=\varphi_{N}1_{\left\\{L_{N}^{2}\|\Psi_{N}^{-1}\|_{\text{op}}^{2}\leq\frac{c_{\eta,T}NT}{\log(NT)}\right\\}}$

An analogous methodology was suggested in [15] within the framework of
classical stochastic differential equations. Nevertheless, the probabilistic
analysis of $\widehat{\varphi}_{N}$ becomes notably more intricate due to the
sophisticated structure of the model. Moreover, unlike the analysis in [15],
the theoretical quantity $\|\Psi^{-1}\|_{\text{op}}^{2}$ in Assumption 4
proves challenging to control due to the absence of information about
$\mu_{t}$. We will discuss this condition in details in Section 3.3.

###  3.2. Main results

This section is devoted to the asymptotic analysis of the estimator
$\widehat{\varphi}_{N}$. We start by introducing two random sets, which will
be crucial for our proofs. We define

(14)
$\Lambda_{N}:=\left\\{L_{N}^{2}\|\Psi_{N}^{-1}\|_{\text{op}}^{2}\leq\frac{c_{\eta,T}NT}{\log(NT)}\right\\},\qquad\Omega_{N}:=\left\\{\left|\frac{\|f\|_{N}^{2}}{\|f\|_{\star}^{2}}-1\right|\leq\frac{1}{2}~{}\forall
f\in S_{N}\right\\}.$

We recall that $\Lambda_{N}$ reflects the cut-off introduced in Eq. 13. On the
other hand, on $\Omega_{N}$ the norms $\|\cdot\|_{N}$ and $\|\cdot\|_{\star}$
are equivalent, i.e.

$\frac{1}{2}\|f\|_{\star}^{2}\leq\|f\|_{N}^{2}1_{\Omega_{N}}\leq\frac{3}{2}\|f\|_{\star}^{2}.$

Our first theoretical statement shows that, under 4, $\mathbb{P}(\Lambda_{N})$
and $\mathbb{P}(\Omega_{N})$ approach $1$ as $N\to\infty$.

######  Proposition 3.1.

Suppose that Assumptions 1 and 4 are satisfied.

  * (i)

There exists $k,n\in\mathbb{N}$ such that for all $N\geq n$:

$\mathbb{P}\left(\left\|\Psi^{-1/2}\Psi_{N}\Psi^{-1/2}-I_{D_{N}}\right\|_{\text{\rm
op}}>\frac{1}{2}\right)\leq D_{N}^{k/4}\exp\left(-\frac{kc_{\eta,T}\eta
NT}{16L_{N}^{2}\|\Psi^{-1}\|_{\text{\rm op}}^{2}}\right),$

where $I_{D_{N}}$ denotes the identity matrix in $\mathbb{R}^{D_{N}\times
D_{N}}$.

  * (ii)

It holds that

$\mathbb{P}(\Lambda_{N}^{c})\leq\mathbb{P}(\Omega_{N}^{c})\leq(NT)^{-\eta+1}.$

The concentration bound in 3.1(i) is key in understanding the asymptotic
behaviour of the estimator $\widehat{\varphi}_{N}$. The main result of this
section is the following theorem.

######  Theorem 3.2.

Suppose that Assumptions 1 and 4 are satisfied. Then we obtain uniform bounds

(15)
$\sup_{\varphi}\mathbb{E}\left[\|\widehat{\varphi}_{N}-\varphi\|_{N}^{2}\right]\leq\sup_{\varphi}\mathbb{E}\left[\inf_{f\in
S_{N}}\|f-\varphi\|_{N}^{2}\right]+\frac{CD_{N}}{NT}$

as well as

$\displaystyle\sup_{\varphi}\mathbb{E}\left[\|\widehat{\varphi}_{N}-\varphi\|_{\star}^{2}\right]$
$\displaystyle\leq\left(1+o(1)\right)\sup_{\varphi}\inf_{f\in
S_{N}}\|f-\varphi\|_{\star}^{2}$ (16)
$\displaystyle+CN^{-1/2}(1+L_{N})\sup_{\varphi}\|\varphi_{\star}-\varphi\|_{\star\star}^{2}+\frac{CD_{N}}{NT},$

where the supremum is taken over the class of functions $\Phi$ defined as

$\Phi:=\left\\{\varphi:\mathbb{R}\to\mathbb{R}:~{}\|\varphi\|_{\infty}\leq
K_{1},~{}K_{2}\leq\|\varphi\|_{\text{\rm Lip}}\leq K_{3}\right\\}$

for fixed constants $K_{1},K_{2},K_{3}>0$. Here the norm $\|f\|_{\star\star}$
is defined as

$\|f\|_{\star\star}^{2}:=\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}f^{2}\star\mu_{t}(x)\mu_{t}(x)\,dx\,dt.$

We remark that $\|f\|_{\star}\leq\|f\|_{\star\star}$. However, it does not
hold a priori that $\|\varphi_{\star}-\varphi\|_{\star\star}\to 0$ as
$N\to\infty$. We will discuss this term in the next subsection.

###  3.3. Checking Assumption 4 and bounding the norm
$\|\varphi_{\star}-\varphi\|_{\star\star}$

The primary challenge in implementing the theoretical findings from the
preceding section lies in selecting an appropriate dimension $D_{N}$, which
ensures the fulfillment of Assumption 4. This requirement is essentially
equivalent to determining an upper bound for the quantity
$\|\Psi^{-1}\|_{\operatorname{op}}$. In this context, we introduce an approach
to establish an upper bound on the operator norm
$\|\Psi^{-1}\|_{\operatorname{op}}$, guided by a Gaussian-type condition on
the densities $\mu_{t}$. We introduce the following condition on
$(\mu_{t})_{t\in[0,T]}$:

######  Assumption 5.

We suppose that the densities $\mu_{t}$ are symmetric and there exist numbers
$c_{1},c_{2}>0$ and functions $g_{2}\in L^{2}([0,T])$, $g_{1}g^{2}_{2}\in
L^{1}([0,T])$ such that

$\mu_{t}(x)\geq
g_{1}(t)\exp(-x^{2}/c_{1})\qquad\text{and}\qquad\widehat{\mu}_{t}(x)\geq
g_{2}(t)\exp(-x^{2}/c_{2}).$

######  Remark 3.3.

The lower bound for the densities $(\mu_{t})_{t\in[0,T]}$ can be deduced via
[37, Theorem 1]. Indeed, under Assumption 2, it holds that

$\displaystyle\mu_{t}(x)$ $\displaystyle\geq\frac{1}{\sqrt{2\pi
t}}\int_{\mathbb{R}}\mu_{0}(y)\int_{\frac{\left|x-y\right|}{\sqrt{t}}}^{\infty}z\exp\left(-\frac{(z+K_{\varphi}\sqrt{t})^{2}}{2}\right)\,dz\,dy$
$\displaystyle\geq\frac{\exp\left(-K_{\varphi}^{2}t\right)}{2\pi\sqrt{t\zeta^{2}}}\int_{\mathbb{R}}\exp\left(-\frac{y^{2}}{2\zeta^{2}}\right)\exp\left(-\frac{(x-y)^{2}}{t}\right)\,dy$
$\displaystyle=\frac{\exp\left(-K_{\varphi}^{2}t\right)}{\sqrt{2\pi(\zeta^{2}+t/2)}}\exp\left(-\frac{x^{2}}{2\zeta^{2}+t}\right).$

Obtaining lower bounds for the Fourier transforms
$(\widehat{\mu}_{t})_{t\in[0,T]}$ is a more delicate problem. Some partial
results in this direction have been demonstrated in [2, Theorem 5.4] in the
ergodic setting. ∎

We consider the vector space $S_{N}$ generated by functions $(e_{k})_{1\leq
k\leq D_{N}}$ with

$e_{k}(x)=(2A_{N})^{-1/2}\exp(i\pi kx/A_{N}),\qquad x\in\mathbb{R},$

which is an orthonormal system in $L^{2}([-A_{N},A_{N}])$. For this choice of
basis we obtain the identity

(17) $\displaystyle\Psi_{kl}=\langle
e_{k},e_{l}\rangle_{\star}=\frac{1}{2A_{N}T}\int_{0}^{T}\widehat{\mu}_{t}(-k/A_{N})\widehat{\mu}_{t}(l/A_{N})\widehat{\mu}_{t}((l-k)/A_{N})dt.$

We deduce the following result.

######  Proposition 3.4.

Suppose that Assumption 5 holds. Assume that $D_{N}=a_{D}\log(N)$ and
$A_{N}=a_{A}\sqrt{\log(N)}$ for some $a_{D},a_{A}>0$. Then it holds that

$\displaystyle\|\Psi^{-1}\|_{\operatorname{op}}\leq
v_{T}N^{\frac{2a_{D}^{2}}{c_{2}a_{A}^{2}}+\frac{a_{A}^{2}}{c_{1}}}(1+o(1)),$

where
$v_{T}:=2\sqrt{2\pi}T\left(\int_{0}^{T}g_{1}(t)g_{2}(t)^{2}dt\right)^{-1}$.

In the next step we will study the norm
$\|\varphi_{\star}-\varphi\|_{\star\star}$. For this purpose, we assume that
$\varphi$ has the representation

$\varphi(x)=\sum_{k=1}^{\infty}c_{k}e_{k}(x)$

with coefficients $c_{k}=c_{k}(N)$ satisfying the condition
$\sum_{k=1}^{\infty}|c_{k}|<\infty$.

######  Proposition 3.5.

Suppose that Assumption 2 is satisfied. Then it holds that

$\|\varphi_{\star}-\varphi\|_{\star\star}^{2}\lesssim\left(1+A_{N}^{-1}D_{N}^{4}\|\Psi^{-1}\|_{\text{\rm
op}}^{2}\exp\left(-\frac{3D_{N}^{2}\zeta^{2}}{2A_{N}^{2}}\right)\right)\left(\sum_{k=D_{N}+1}^{\infty}|c_{k}|\right)^{2}.$

Now, we can combine the statements of Theorem 3.2, Proposition 3.4 and
Proposition 3.5, to derive the convergence rate for
$\|\widehat{\varphi}_{N}-\varphi\|_{\star}$.

For clarity, we focus on the ergodic scenario where $\mu_{t}=\mu_{0}$ and
$\mu_{0}$ satisfies Assumption 2. In this context, the constants $c_{1}$ and
$c_{2}$ from Assumption 5 are explicitly given by

$c_{1}=2\zeta^{2},\qquad c_{2}=\frac{2}{\zeta^{2}}.$

As $\|e_{j}\|_{\infty}^{2}=A_{N}^{-1}$ and $D_{N}=a_{D}\log(N)$,
$A_{N}=a_{A}\sqrt{\log(N)}$, Proposition 3.4 implies the following bound:

$L_{N}^{2}\|\Psi^{-1}\|_{\text{op}}^{2}\lesssim\frac{a_{D}^{2}}{a_{A}^{2}}\log(N)N^{\frac{2a_{D}^{2}\zeta^{2}}{a_{A}^{2}}+\frac{a_{A}^{2}}{\zeta^{2}}}.$

Hence, for Assumption 4 to hold true, the constants $a_{A}$ and $a_{D}$ need
to fulfill

$\frac{2a_{D}^{2}\zeta^{2}}{a_{A}^{2}}+\frac{a_{A}^{2}}{\zeta^{2}}<1.$

On the other hand, exploring the proof of Theorem 2.6, the approximation error
is obtained as

$\|\varphi_{\star}-\varphi\|_{\star}^{2}\lesssim
N^{-\frac{a_{D}^{2}\zeta^{2}}{a_{A}^{2}}}+N^{-\frac{a_{A}^{2}}{2\zeta^{2}}}.$

Consequently, if we choose
$a_{D}^{2}\zeta^{2}/a_{A}^{2}=a_{A}^{2}/(2\zeta^{2})=(1-\epsilon)/4$ for a
small $\epsilon>0$, we finally conclude that

(18)
$\displaystyle\mathbb{E}\left[\|\widehat{\varphi}_{N}-\varphi\|_{\star}^{2}\right]\lesssim
N^{-1/4+\epsilon}$

provided the condition $(\sum_{k=D_{N}+1}^{\infty}|c_{k}|)^{2}\leq CN^{-z}$
with $z>5/8$ holds. The substantial decrease in the convergence rate, as
observed compared to Theorem 2.6, directly stems from the stringent constraint
imposed by Assumption 4. Still this is the first result in literature
regarding the convergence rates of the linear-type estimates within the
context of McKean-Vlasov Stochastic Differential Equations.

##  4\. Lower bounds

This section is dedicated to deriving minimax lower bounds. We start by
establishing a lower bound for the previously examined estimation problem
concerning the norm $\|\cdot\|_{\star}$. To achieve this, we focus on a
simplified scenario involving i.i.d. observations drawn from a McKean-Vlasov
SDE:

$\left\\{\begin{array}[]{ll}dX_{t}=(\varphi\star\mu_{t})(X_{t})\,dt+\sigma
dB_{t},\\\ \textnormal{Law}(X_{t})=\mu_{t},\end{array}\right.$

and denote by $\mathbb{P}_{\varphi}$ the associated probability measure. We
assume that $X_{0}^{\varphi}=x\in\mathbb{R}$ for all $\varphi$. The ensuing
theorem establishes a minimax lower bound with respect to the norm
$\|\cdot\|_{\star}$.

######  Theorem 4.1.

There exists a constant $c>0$ such that, for every $N\in\mathbb{N}$,

(19) $\inf_{\widehat{\varphi}_{N}}\sup_{\varphi}\mathbb{P}^{\bigotimes
N}_{\varphi}\left(\|\widehat{\varphi}_{N}-\varphi\|_{\star}>cN^{-1/2}\right)>0,$

where $\mathbb{P}^{\bigotimes N}_{\varphi}$ is the $N$-fold product measure,
the supremum is taken over all functions $\varphi$ satisfying Assumption 3 and
$\|\varphi\|_{\infty}\leq(32T)^{-1/2}$, and the infimum is taken over all
estimators of $\varphi$ retrieved from $N$ observations of
$(X_{t})_{t\in[0,T]}$.

###### Proof.

Consider two functions $\varphi_{0},\varphi_{1}\in S$ and denote by
$\mu_{t}^{0}$ and $\mu_{t}^{1}$ the corresponding marginal densities. The
Kullback-Leibler divergence
$\text{KL}(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}})$ between the
probability measures $\mathbb{P}_{\varphi_{0}}$ and $\mathbb{P}_{\varphi_{1}}$
can be explicitly computed as

$\displaystyle\text{KL}(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}})$
$\displaystyle=\frac{1}{2}\int_{0}^{T}\int_{\mathbb{R}}\left(\varphi_{0}\star\mu_{t}^{0}(x)-\varphi_{1}\star\mu_{t}^{1}(x)\right)^{2}\mu_{t}^{0}(x)\,dx\,dt.$

As a consequence we obtain the inequality

$\displaystyle\text{KL}(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}})$
$\displaystyle\leq
T\|\varphi_{0}-\varphi_{1}\|_{\star}^{2}+\|\varphi_{1}\|_{\infty}^{2}\int_{0}^{T}\left[\int_{\mathbb{R}}\left|\mu_{t}^{0}(x)-\mu_{t}^{1}(x)\right|\,dx\right]^{2}\,dt,$

where the $\|\cdot\|_{\star}$-norm is computed with respect to $\mu_{t}^{0}$.
Furthermore, using Theorem 1.1 of [8], we derive

$\displaystyle\int_{0}^{T}\left[\int_{\mathbb{R}}\left|\mu_{t}^{0}(x)-\mu_{t}^{1}(x)\right|\,dx\right]^{2}dt$
$\displaystyle\leq
4\int_{0}^{T}\left[\int_{0}^{t}\int(\varphi_{0}\star\mu_{s}^{0}(x)-\varphi_{1}\star\mu_{s}^{1}(x))^{2}\mu_{s}^{0}(x)\,dx\,ds\right]\,dt$
$\displaystyle\leq
8T\text{KL}(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}}).$

Since $\|\varphi_{1}\|_{\infty}^{2}\leq(16T)^{-1}$ we finally deduce that

$\text{KL}(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}})\leq
2T\|\varphi_{0}-\varphi_{1}\|_{\star}^{2}.$

Applying the two hypotheses method described in [42, Theorem 2.2], we obtain
the statement of the theorem. ∎

We will now consider the $L^{2}(\mathbb{R})$-norm instead of
$\|\cdot\|_{\star}$ and compare the results to [5]. We recall that logarithmic
rates have been obtained in the $L^{2}(\mathbb{R})$-norm in [5, Theorem 5.1]
for a similar estimation problem given the observation scheme
$(X_{T}^{i,N})_{1\leq i\leq N}$ with $N,T\to\infty$. It turns out that these
rates cannot be improved to polynomial ones in their framework, and the same
holds in our case (note however that polynomial rates may appear under a
different set of conditions as it has been demonstrated in [2]). To see this,
we again consider simplified setting of i.i.d. observations drawn from McKean-
Vlasov SDE Eq. 3. Furthermore, we consider a similar model as presented in
[5]: Let $W:\mathbb{R}\to\mathbb{R}$ be an even and strictly convex
interaction potential of the form

$W(x)=p(x)+\beta(x),\qquad p(x)=\sum_{j=1}^{J}a_{j}x^{2j},$

where $(a_{j})_{j=1,\dots,J}$ are known constants and $\beta\in
C_{c}^{2}(\mathbb{R})$. The potential $W$ determines the interaction function
$\varphi$ via $\varphi=W^{\prime}$ and we are interested in the estimation of
the non-parametric part $\beta$ in the case of stationarity. The invariant law
$\mu$ solves the elliptic nonlinear Fokker-Planck equation

(20)
$\left[\frac{1}{2}\frac{\textnormal{d}^{2}}{\textnormal{d}x^{2}}+\frac{\textnormal{d}}{\textnormal{d}x}\left(\varphi\star\mu\right)\right]\mu=0,$

meaning that $\mu$ is given by the implicit equation

(21) $\mu(x)=c_{\mu}\exp\left(-W\star\mu(x)\right),$

where $c_{\mu}$ is a normalising constant such that $\mu$ is a probability
density.

To introduce the appropriate class of functions, we consider
$C_{c}^{2}(\mathbb{R})$-wavelets. More specifically, we let the mother wavelet
be a symmetric function $\psi\in C_{c}^{2}(\mathbb{R})$ with
$\text{supp}(\psi)=[-1/2,1/2]$ and, for $n,k\in\mathbb{Z}$, define

$\psi_{n,k}(x)=2^{n/2}\psi(2^{n}x-k).$

The set $(\psi_{n,k})_{n,k\in\mathbb{Z}}$ forms a basis of
$L^{2}(\mathbb{R})$. Now, for $\alpha,\ K>0$, we consider the functional space

$\mathcal{F}^{\alpha}_{K}=\left\\{\beta^{\prime}:~{}\beta\in
C^{2}_{c}(\mathbb{R}),\
\sum_{n=1}^{\infty}\sum_{|k|>m}\langle\beta^{\prime},\psi_{n,k}\rangle_{L^{2}(\mathbb{R})}^{2}\leq
Km^{-\alpha}\quad\forall m\in\mathbb{N}\right\\}.$

For the estimation of the nonparametric part $\beta^{\prime}$ of $\varphi$
based on i.i.d. observations of Eq. 3 we obtain the following lower bound.

######  Theorem 4.2.

Assume that the coefficients $(a_{j})_{1\leq j\leq J}$ and the function
$\beta$ satisfy the conditions of [5, Theorem 5.1]. Then there exists a
constant $c>0$ such that, for every $N\in\mathbb{N}$,

(22)
$\inf_{\widehat{\beta}^{\prime}_{N}}\sup_{\beta^{\prime}\in\mathcal{F}^{\alpha}_{K}}\mathbb{P}^{\bigotimes
N}_{\varphi}\left(\|\widehat{\beta}^{\prime}_{N}-\beta^{\prime}\|_{L^{2}(\mathbb{R})}^{2}>c\left(\log
N\right)^{-\frac{\alpha}{4J}}\right)>0,$

where $\mathbb{P}^{\bigotimes N}_{\varphi}$ is the $N$-fold product measure
and the infimum is taken over all estimators of $\beta^{\prime}$ retrieved
from $N$ observations of $(X_{t})_{t\in[0,T]}$.

###### Proof.

We will use the two hypotheses method described in [42, Theorem 2.2]. We will
find two hypotheses of interaction forces
$\varphi_{0},\varphi_{1}\in\mathcal{F}^{\alpha}_{K}$ such that, for some
constant $c>0$,

$\left\|\varphi_{0}-\varphi_{1}\right\|_{L^{2}(\mathbb{R})}^{2}=c\left(\log
N\right)^{-\frac{\alpha}{4J}}\quad\text{and}\quad\text{KL}\left(\mathbb{P}^{\bigotimes
N}_{\varphi_{0}},\mathbb{P}^{\bigotimes N}_{\varphi_{1}}\right)\lesssim 1.$

The Radon-Nikodym density of
$\textnormal{d}\mathbb{P}_{\varphi_{1}}/\textnormal{d}\mathbb{P}_{\varphi_{0}}$
can be computed using Girsanov’s theorem: denoting the stationary marginals as
$\mu_{i}$, $i=0,1$, we have for $X\in C([0,T];\mathbb{R})$

$\displaystyle\log\left(\frac{\textnormal{d}\mathbb{P}_{\varphi_{1}}}{\textnormal{d}\mathbb{P}_{\varphi_{0}}}(X)\right)$
$\displaystyle=\int_{0}^{T}\varphi_{1}\star\mu_{1}(X_{t})-\varphi_{0}\star\mu_{0}(X_{t})\textnormal{d}X_{t}$
$\displaystyle-\frac{1}{2}\int_{0}^{T}\varphi_{1}\star\mu_{1}(X_{t})^{2}-\varphi_{0}\star\mu_{0}(X_{t})^{2}\textnormal{d}t,$

such that

$\displaystyle\text{KL}\left(\mathbb{P}_{\varphi_{0}},\mathbb{P}_{\varphi_{1}}\right)$
$\displaystyle=\int_{\mathbb{R}}\log\left(\frac{\textnormal{d}\mathbb{P}_{\varphi_{1}}}{\textnormal{d}\mathbb{P}_{\varphi_{0}}}(x)\right)\mu_{1}(x)\textnormal{d}x$
$\displaystyle=\frac{T}{2}\int_{\mathbb{R}}\left(\varphi_{1}\star\mu_{1}(x)-\varphi_{0}\star\mu_{0}(x)\right)^{2}\mu_{1}(x)\textnormal{d}x.$

In a similar manner as in [5, Theorem 5.1], we can show that

$\text{KL}\left(\mathbb{P}_{\varphi_{1}},\mathbb{P}_{\varphi_{0}}\right)\leq\gamma^{2}\text{KL}\left(\mathbb{P}_{\varphi_{1}},\mathbb{P}_{\varphi_{0}}\right)+\frac{T}{2}\int_{\mathbb{R}}(\beta^{\prime}_{0}-\beta^{\prime}_{1})\star\mu_{0}(x)^{2}\mu_{1}(x)\textnormal{d}x,$

with $\gamma\in(0,1)$. So, it remains to show the estimate

(23)
$\int_{\mathbb{R}}(\beta^{\prime}_{0}-\beta^{\prime}_{1})\star\mu_{0}(x)^{2}\mu_{1}(x)\textnormal{d}x\lesssim\frac{1}{N}.$

Now we will choose the two hypotheses. Let $\rho>0$, $M\in\mathbb{N}$ be
rescaling parameters to be determined later. For $n\in\mathbb{N}$, we write
$l_{n}:=\lfloor\log_{2}n\rfloor$ and define

$f=\rho\sum_{\left|n\right|,\left|k\right|=M}^{2M}\psi_{l_{n},k}.$

Note that our choice of basis elements implies that
$\operatorname{supp}f\subset[-2M^{2},2M^{2}]$ and
$\left\|f\right\|_{L^{2}(\mathbb{R})}^{2}=\rho^{2}M^{2}$. We will consider the
hypotheses $\varphi_{0}\in\mathcal{F}^{\alpha}_{K}$ and
$\varphi_{1}=\varphi_{0}+f$. In order for
$\varphi_{1}\in\mathcal{F}^{\alpha}_{K}$ to hold, we need $d(f,S_{m})\leq
Km^{-\alpha}$ for all $m\in\mathbb{N}$. There are two cases to consider for
this, namely

$\rho^{2}\sum_{n_{1}=1}^{\infty}\sum_{\left|k_{1}\right|>m}\sum_{\left|n_{2}\right|,\left|k_{2}\right|=M}^{2M}\left<\psi_{n_{1},k_{1}},\psi_{{l_{n_{2}}},k_{2}}\right>_{2}^{2}\begin{cases}\lesssim\left\|f\right\|_{2}^{2}=\rho^{2}M^{2},&m\leq
l_{2M},\\\ =0,&m>l_{2M}.\end{cases}$

Hence, we require the condition
$\rho^{2}M^{2}\lesssim\lfloor\log_{2}(2M)\rfloor^{-\alpha}$. We enforce a
stricter condition

$\rho^{2}=O(M^{-\alpha-2}).$

This means we choose

$\left\|\varphi_{0}-\varphi_{1}\right\|_{L^{2}(\mathbb{R})}^{2}=\left\|f\right\|_{L^{2}(\mathbb{R})}^{2}=\rho^{2}M^{2}=M^{-\alpha}.$

We will now verify that Eq. 23 holds. The main ingredient is the observation
that, for $i=0,1$,

(24) $\mu_{i}(x)\lesssim\exp\left(-a_{J}x^{2J}\right),$

which in particular implies $\|\mu_{i}\|_{L^{2}(\mathbb{R})}<\infty$. By
symmetry of $\mu_{i}$ and $f$, it suffices to show

(25)
$\int_{0}^{\infty}f\star\mu_{0}(x)^{2}\mu_{1}(x)\textnormal{d}x\lesssim\frac{1}{N}.$

Let us split the integral at some $k>0$ which we determine later. The tail
behaviour is governed by Eq. 24: Using Cauchy-Schwarz, we have
$\|f\star\mu_{0}\|_{\infty}^{2}\leq\|f\|_{L^{2}(\mathbb{R})}^{2}\|\mu_{0}\|_{L^{2}(\mathbb{R})}^{2}\lesssim\rho^{2}M^{2}$,
such that

$\int_{k}^{\infty}f\star\mu_{0}(x)^{2}\mu_{1}(x)\textnormal{d}x\lesssim\rho^{2}M^{2}\frac{\exp\left(-k^{2J}\right)}{k^{2J-1}}.$

For the remaining integral, we bound $\mu_{1}$ such that

$\int_{0}^{k}f\star\mu_{0}(x)^{2}\mu_{1}(x)\lesssim
k\sup_{x\in[0,k]}f\star\mu_{0}(x)^{2}.$

We recall that $\operatorname{supp}f=[-2M^{2},2M^{2}]$, which implies

$\displaystyle\sup_{x\in[0,k]}\int_{-\infty}^{x-2M^{2}}f(x-y)\mu_{0}(y)\textnormal{d}y$
$\displaystyle\leq\left\|f\right\|_{L^{2}(\mathbb{R})}^{2}\int_{-\infty}^{k-2M^{2}}\mu_{0}(y)^{2}\textnormal{d}y$
$\displaystyle\lesssim\rho^{2}M^{2}\frac{\exp\left(-(2M^{2}-k)^{2J}\right)}{(2M^{2}-k)^{2J-1}}.$

From this, we see that by choosing $k=M^{2}$, the entire integral is indeed
bounded by

$\int_{0}^{\infty}f\star\mu_{0}(x)^{2}\mu_{1}(x)\textnormal{d}x\lesssim\exp\left(-M^{4J}\right),$

such that choosing $M=c\left(\log N\right)^{\frac{1}{4J}}$ with some constant
$c>0$ gives Eq. 25, which in turn gives us the rate

$\left\|\varphi_{0}-\varphi_{1}\right\|_{L^{2}(\mathbb{R})}^{2}=CM^{-\alpha}=C\left(\log
N\right)^{-\frac{\alpha}{4J}},$

which implies the claim of the theorem. ∎

##  5\. Proofs

This section is devoted to proofs of the main results. We will often transfer
probabilistic results from i.i.d. observations of the McKean-Vlasov SDE
displayed in Eq. 3 to observations of the original particle system at Eq. 1
via a change of measure device established in [17]. For this purpose we
consider two $N$-dimensional processes given by the weak solutions of

$\begin{array}[]{cc}d\overline{X}_{t}^{i}&=(\varphi\star\mu_{t})(\overline{X}_{t}^{i})+\sigma
dW_{t}^{i}\end{array},\quad 1\leq i\leq N,$
$\begin{array}[]{cc}dX_{t}^{i}&=(\varphi\star\mu_{t}^{N})(X_{t}^{i})+\sigma
dW_{t}^{i}\end{array},\quad 1\leq i\leq N$

having the same initial value, and denote by $\overline{\mathbb{P}}^{N}$ and
$\mathbb{P}^{N}$, respectively, the associated probability measures.

###  5.1. Proof of Theorem 2.3

In the first part of the proof we consider the setting of i.i.d. observations
drawn from the mean field limit Eq. 3, that is we work under
$\overline{\mathbb{P}}^{N}$. We recall the inequality

(26)
$\|\varphi_{N}-\varphi\|_{\star}^{2}\leq\|\varphi_{\star}-\varphi\|_{\star}^{2}+2\left(\sup_{f\in
S_{N}}\left|\|f-\varphi\|_{N}^{2}-\|f-\varphi\|_{\star}^{2}\right|+\sup_{f\in
S_{N}}|\gamma^{\prime}_{N}(f)|\right),$

where $\varphi_{\star}$ has been introduced in Eq. 7. Next, we set
$\gamma^{\prime}_{N}(f)=:2M_{N}(f)/\sqrt{NT}$ where for all $f\in S$,
$M_{N}(f)$ is a martingale with quadratic variation

$\frac{\sigma^{2}}{NT}\sum_{i=1}^{N}\int_{0}^{T}[(f\star\mu_{t}^{N})(X_{t}^{i,N})]^{2}dt\leq\sigma^{2}\|f\|^{2}_{\infty}.$

Hence, we deduce by the Bernstein inequality for continuous martingales,

$\overline{\mathbb{P}}^{N}(|M_{N}(f)-M_{N}(g)|\geq a)\leq
2\exp\left(-a^{2}/(\sigma^{2}\|f-g\|^{2}_{\infty})\right)$

for any $a>0$ and $f,g\in S_{N}$. Since for all $q\geq 1$, the
$L^{q}(\Omega)$-norm is dominated by $\left\|\cdot\right\|_{\psi_{e,2}}$, an
application of the general maximal inequality [32, Theorem 8.4] implies

(27) $\displaystyle\Bigl{\|}\sup_{f\in
S_{N}}\left|M_{N}(f)-M_{N}(f_{0})\right|\Bigr{\|}_{L^{2}(\Omega)}\lesssim\mathsf{DI}(S_{N},\|\cdot\|_{\infty},\psi_{e,2})$

for any $f_{0}\in S_{N}$. Consider now the difference
$\|g\|_{N}^{2}-\|g\|_{\star}^{2}$ for a function $g\in S_{N}(2K_{\varphi})$.
We obtain the decomposition

(28)
$\displaystyle\|g\|_{N}^{2}-\|g\|_{\star}^{2}=U_{1,N}(g)+U_{2,N}(g)+U_{3,N}(g)+O(N^{-1})$

where $\Delta_{t}(x,y):=g(x-y)-(g\star\mu_{t})(x),$ and

$\displaystyle U_{1,N}(g)$
$\displaystyle:=\frac{2}{T}\int_{0}^{T}\int_{\mathbb{R}}\left(\frac{1}{N}\sum_{j=1}^{N}\Delta_{t}(x,X_{t}^{j,N})\right)(g\star\mu_{t})(x)\mu_{t}(x)dxdt$
$\displaystyle+\left(\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}(g\star\mu_{t})(X_{t}^{i,N})^{2}dt-\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}(g\star\mu_{t})(x)^{2}\mu_{t}(x)dxdt\right),$
$\displaystyle U_{2,N}(g)$
$\displaystyle:=\frac{1}{N(N-1)(N-2)}\sum_{i=1}^{N}\sum_{j,k\not=i,j\not=k}\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}\Delta_{t}(x,X_{t}^{j,N})\Delta_{t}(x,X_{t}^{k,N})\mu_{t}(x)dxdt$
$\displaystyle+\frac{2}{NT}\sum_{i=1}^{N}\int_{0}^{T}\frac{1}{N-1}\sum_{j\not=i}\Delta_{t}(X_{t}^{i,N},X_{t}^{j,N})(g\star\mu_{t})(X_{t}^{i,N})dt$
$\displaystyle-\frac{2}{NT}\sum_{i=1}^{N}\int_{0}^{T}\frac{1}{N-1}\sum_{j\not=i}\int_{\mathbb{R}}\Delta_{t}(x,X_{t}^{j,N})(g\star\mu_{t})(x)\mu_{t}(x)\,dx\,dt,$
$\displaystyle U_{3,N}(g)$
$\displaystyle:=\frac{1}{N(N-1)(N-2)}\sum_{i=1}^{N}\sum_{j,k\not=i,j\not=k}\frac{1}{T}\int_{0}^{T}\Delta_{t}(X_{t}^{i,N},X_{t}^{j,N})\Delta_{t}(X_{t}^{i,N},X_{t}^{k,N})\,dt$
$\displaystyle-\frac{1}{N(N-1)(N-2)}\sum_{i=1}^{N}\sum_{j,k\not=i,j\not=k}\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}\Delta_{t}(x,X_{t}^{j,N})\Delta_{t}(x,X_{t}^{k,N})\mu_{t}(x)\,dx\,dt.$

The key observation is that each $U_{r,N}$ is a degenerate $U$-statistics of
order $r$ for $r=1,2,3$, while $U_{1,N}$ is the leading term. First, turn to
$U_{1,N}$ and note that

$\displaystyle
U_{1,N}(g)=\frac{1}{N}\sum_{j=1}^{N}\left(Z_{j}(g)-\mathbb{E}_{\overline{\mathbb{P}}^{N}}Z_{j}(g)\right)$

with

$\displaystyle
Z_{j}(g)=\frac{1}{T}\int_{0}^{T}\left(2\int_{\mathbb{R}}\Delta_{t}(x,X_{t}^{j,N})(g\star\mu_{t})(x)\mu_{t}(x)\,dx\right)+(g\star\mu_{t})(X_{t}^{i,N})^{2}\,dt.$

It holds that

$|Z_{j}(g)|\leq 8K_{\varphi}^{2},\quad|Z_{j}(g)-Z_{j}(g^{\prime})|\leq
4K_{\varphi}\|g-g^{\prime}\|_{\infty},$

with probability $1$ for $g,g^{\prime}\in S_{N}(2K_{\varphi}).$ Hence,

$\|Z_{j}(g)-Z_{j}(g^{\prime})\|_{\psi_{e,2}}\lesssim
K_{\varphi}\|g-g^{\prime}\|_{\infty}.$

This implies that the process $\widetilde{U}_{1,N}=\sqrt{N}U_{1,N}$ has
subgaussian increments and

$\displaystyle\|\widetilde{U}_{1,N}(g)-\widetilde{U}_{1,N}(g^{\prime})\|_{\psi_{e,2}}\lesssim
K_{\varphi}\|g-g^{\prime}\|_{\infty}.$

Fix some $g_{0}\in S_{N}(2K_{\varphi})$. By the triangle inequality,

$\sup_{g\in
S_{N}(2K_{\varphi})}|\widetilde{U}_{1,N}(g)|\leq\sup_{g,g^{\prime}\in
S_{N}(2K_{\varphi})}|\widetilde{U}_{1,N}(g)-\widetilde{U}_{1,N}(g^{\prime})|+|\widetilde{U}_{1,N}(g_{0})|.$

By the Dudley integral inequality, see [43, Theorem 8.1.6], for any
$\delta\in(0,1)$, we conclude that

$\sup_{g,g^{\prime}\in
S_{N}(2K_{\varphi})}|\widetilde{U}_{1,N}(g)-\widetilde{U}_{1,N}(g^{\prime})|\lesssim
K_{\varphi}\bigl{[}\mathsf{DI}(S_{N},\|\cdot\|_{\infty},\psi_{e,2})+\sqrt{\log(2/\delta)}\bigr{]}$

holds with probability at least $1-\delta$. Applying Hoeffding’s inequality,
see e.g. [43, Theorem 2.6.2.], we have for any $\delta\in(0,1)$,

$|\widetilde{U}_{1,N}(g_{0})|\lesssim K_{\varphi}\sqrt{\log(1/\delta)},$

with probability at least $1-\delta$. In other words, for any $u>0$,

$\overline{\mathbb{P}}^{N}\left(\sup_{g\in
S_{N}(2K_{\varphi})}|\widetilde{U}_{1,N}(g)|>u\right)\leq\exp\left(-\frac{(u-K_{\varphi}\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{e,2}))^{2}}{2K_{\varphi}^{2}}\right).$

Putting things together we conclude that

(29)
$\displaystyle\Bigl{\\{}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\Bigl{[}\sup_{g\in
S_{N}(2K_{\varphi})}|U_{1,N}(g)|^{q}\Bigr{]}\Bigr{\\}}^{1/q}\lesssim\frac{qK_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{e,2})}{\sqrt{N}}.$

Furthermore, due to Theorem A.1,

(30)
$\displaystyle\Bigl{\\{}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\Bigl{[}\sup_{g\in
S_{N}(2K_{\varphi})}\left|NU_{2,N}(g)\right|^{q}\Bigr{]}\Bigr{\\}}^{1/q}\lesssim
p^{2}K_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{p/2})$

and

(31)
$\displaystyle\Bigl{\\{}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\Bigl{[}\sup_{g\in
S_{N}(2K_{\varphi})}\left|N^{3/2}U_{3,N}(g)\right|^{q}\Bigr{]}\Bigr{\\}}^{1/q}\lesssim
p^{3}K_{\varphi}\,\mathsf{DI}(S_{N}(2K_{\varphi}),\|\cdot\|_{\infty},\psi_{p/3})$

for any $p>2$ and $q\leq p.$ This implies the statement of Theorem 2.3 in the
setting of i.i.d. observations drawn from the McKean-Vlasov SDE Eq. 3.

Now, we will transfer the result from i.i.d. observations to observations of
the original particle system at Eq. 1 via a change of measure device
established in [17]. Recalling the definition of probability measures
$\overline{\mathbb{P}}^{N}$ and $\mathbb{P}^{N}$, we deduce the identity

$\frac{\textnormal{d}\mathbb{P}^{N}}{\textnormal{d}\overline{\mathbb{P}}^{N}}=\mathcal{E}_{T}(\overline{M}^{N})$

where
$\mathcal{E}_{t}(\overline{M}^{N})=\exp\bigl{(}\overline{M}_{t}^{N}-\frac{1}{2}\langle\overline{M}^{N}\rangle_{t}\bigr{)}$
and

$\overline{M}_{t}=\sum_{i=1}^{N}\int_{0}^{t}\sigma^{-1}\left[\varphi\star(\mu_{t}^{N}-\mu_{t})(\overline{X}_{t}^{i})\right]\,dW_{t}^{i}$

is a $\overline{\mathbb{P}}^{N}$-local martingale. According to [17,
Proposition 19], we obtain the bound

(32) $\sup_{N\geq
1}\sup_{t\in[0,T-\delta]}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[\exp\left(\tau\left(\langle\overline{M}^{N}\rangle_{t+\delta}-\langle\overline{M}^{N}\rangle_{t}\right)\right)\right]\leq
C_{\delta,\tau}$

for every $\tau>0,$ $0\leq\delta\leq\delta_{0}$ and some constant
$C_{\delta,\tau}>0$. Let $\xi$ be any $\mathcal{F}_{T}$-measurable random
variable. Then by the Hölder inequality

(33)
$\displaystyle\mathbb{E}_{\mathbb{P}^{N}}[|\xi|^{r}]=\mathbb{E}_{\overline{\mathbb{P}}^{N}}[|\xi|^{r}\mathcal{E}_{T}(\overline{M}^{N})]\leq\left\\{\mathbb{E}_{\overline{\mathbb{P}}^{N}}[|\xi|^{2r}]\right\\}^{1/2}\left\\{\mathbb{E}_{\overline{\mathbb{P}}^{N}}[\mathcal{E}_{T}^{2}(\overline{M}^{N})]\right\\}^{1/2}.$

Next, fix a grid $0=t_{0}<t_{1}<\ldots<t_{K}=T$ with
$|t_{k+1}-t_{k}|\leq\delta$ and decompose

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}[\mathcal{E}_{T}^{2}(\overline{M}^{N})]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\prod_{k=1}^{K}\exp\Bigl{(}2(\overline{M}_{t_{k}}^{N}-\overline{M}_{t_{k-1}}^{N})+\langle\overline{M}^{N}\rangle_{t_{k-1}}-\langle\overline{M}^{N}\rangle_{t_{k}}\Bigr{)}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\prod_{k=1}^{K}\mathcal{E}_{t_{k}}(\overline{M}_{t_{k}}^{N}-\overline{M}_{t_{k-1}}^{N})\exp\Bigl{(}\langle\overline{M}_{t_{k}}^{N}-\overline{M}_{t_{k-1}}^{N}\rangle$
$\displaystyle+$
$\displaystyle\langle\overline{M}^{N}\rangle_{t_{k-1}}-\langle\overline{M}^{N}\rangle_{t_{k}}\Bigr{)}$

By using martingale property of $\mathcal{E}_{t_{k}},$ repeatedly applying the
Hölder inequality and using (32), we deduce the inequality

$\mathbb{E}_{\overline{\mathbb{P}}^{N}}[\mathcal{E}_{T}^{2}(\overline{M}^{N})]\leq
C_{\delta,T}.$

As a consequence the bounds Eq. 27-Eq. 31 remain valid under the probability
measure $\mathbb{P}^{N}$. This completes the proof.

###  5.2. Proof of Theorem 2.6

Note that $\alpha_{n_{N}-1}<A_{N}\leq\alpha_{n_{N}}$ and hence
$A_{N}/\alpha_{n_{N}}\to 1$ as $N\to\infty.$ Under Assumption 3 considered for
the sequence $\alpha_{n_{N}}$, we obtain the inequality

$\displaystyle\inf_{g\in S_{N}}$
$\displaystyle\frac{1}{T}\int_{0}^{T}\int_{\mathbb{R}}\left((\varphi\star\mu_{t})(x)-(g\star\mu_{t})(x)\right)^{2}\mu_{t}(x)\,dx\,dt$
$\displaystyle\leq\frac{1}{\alpha_{n_{N}}T}\int_{0}^{T}\|\mu_{t}\|_{[-\alpha_{n_{N}},\alpha_{n_{N}}]}\int_{-\alpha_{n_{N}}}^{\alpha_{n_{N}}}\left|\sum_{k=D_{N}+1}^{\infty}c_{k}\exp(i\pi
kx/\alpha_{n_{N}})\widehat{\mu}_{t}(-k/\alpha_{n_{N}})\right|^{2}\,dx\,dt+R(\alpha_{n_{N}})$
$\displaystyle\leq\frac{1}{\alpha_{n_{N}}T}\int_{0}^{T}\|\mu_{t}\|_{[-\alpha_{n_{N}},\alpha_{n_{N}}]}\sum_{k=D_{N}+1}^{\infty}\left(|c_{k}||\widehat{\mu}_{t}(-k/\alpha_{n_{N}})|\right)^{2}\,dt+R(\alpha_{n_{N}})$

where

$\displaystyle
R(A)\leq\frac{4K_{\varphi}^{2}}{T}\int_{0}^{T}\int_{|x|>A}\mu_{t}(x)\,dx\,dt.$

In the next step we would like to obtain estimates for the marginal densities
$\mu_{t}$. For this purpose we consider the SDE

(34) $\displaystyle
d\widetilde{X}_{t}=(\varphi\star\mu_{t})(\widetilde{X}_{t})\,dt+\sigma
dB_{t},\quad\widetilde{X}_{t}\sim\mu_{0}$

which is obtained by freezing the densities $\mu_{t}$ in the McKean-Vlasov
equation Eq. 3. Note that, under Assumption 1, (34) is the standard SDE with
uniformly bounded (by $K_{\varphi}$) and uniformly Lipschitz continuous drift
function $\varphi\star\mu_{t}.$ By applying [41, Theorem 3.1] to (3) and using
our Assumption 2, we deduce that

$\displaystyle\mu_{t}(x)$ $\displaystyle\leq\frac{1}{\sqrt{2\pi
t}}\int_{\mathbb{R}}\mu_{0}(y)\int_{\frac{\left|x-y\right|}{\sqrt{t}}}^{\infty}z\exp\left(-\frac{(z-K_{\varphi}\sqrt{t})^{2}}{2}\right)dzdy$
$\displaystyle\leq\frac{\exp\left(K_{\varphi}^{2}T/2\right)}{2\pi\sqrt{t\zeta^{2}}}\int_{\mathbb{R}}\exp\left(-\frac{y^{2}}{2\zeta^{2}}\right)\exp\left(-\frac{(x-y)^{2}}{2t}\right)dy$
$\displaystyle=\frac{\exp\left(K_{\varphi}^{2}T/2\right)}{\sqrt{2\pi(\zeta^{2}+t)}}\exp\left(-\frac{x^{2}}{2(\zeta^{2}+t)}\right).$

Therefore, we get

$\displaystyle R(A)\leq
8K_{\varphi}^{2}\exp\left(K_{\varphi}^{2}T/2\right)\exp\left(-\frac{A^{2}}{2(\zeta^{2}+T)}\right).$

Next, note that the equation (3) can be written as

$\displaystyle X_{t}=X_{0}+\int_{0}^{t}(\varphi\star\mu_{s})(X_{s})\,ds+\sigma
B_{t}$

meaning that $\widehat{\mu}_{t}(u)=\widehat{\mu}_{0}(u)\psi_{t}(u)$ where
$\psi_{t}$ is the characteristic function of the random variable
$\int_{0}^{t}(\varphi\star\mu_{s})(X_{s})\,ds+\sigma B_{t}.$ Thus, under
Assumption 2, we conclude that

(35) $\displaystyle|\widehat{\mu}_{t}(u)|\leq\exp(-u^{2}\zeta^{2}/2).$

Furthermore, by [41, Remark 3.3], $\|\mu_{t}\|_{\infty}\leq
2\|\mu_{0}\|_{\infty}+K_{\varphi}.$ As a consequence we derive

$\displaystyle\inf_{g\in S_{N}}\left\|g-\varphi\right\|_{\star}^{2}$
$\displaystyle\lesssim\frac{K_{\varphi}}{A_{N}}\sum_{k=D_{N}+1}^{\infty}|c_{k}|^{2}\exp\left(-\frac{k^{2}\zeta^{2}}{A_{N}^{2}}\right)$
$\displaystyle+\exp\left(K_{\varphi}^{2}T/2\right)\exp\left(-\frac{A_{N}^{2}}{2(\zeta^{2}+T)}\right)$

Hence under our choice of $A_{N}$ and $D_{N}$ we obtain that

$\displaystyle\inf_{g\in
S_{N}}\|g-\varphi\|_{\star}^{2}\lesssim\frac{\exp(K_{\varphi}^{2}T/2)}{N}.$

This completes the proof of Theorem 2.6.

###  5.3. Proof of Proposition 3.1

First, we present some preliminary results.

####  5.3.1. Some properties of $\Psi$

We begin by pointing out that $\Psi$ is invertible. Indeed, this immediately
follows from the representation $\Psi=(\langle
e_{i},e_{j}\rangle_{\star})_{1\leq i,j\leq D_{N}}$, since $(e_{j})_{1\leq
j\leq D_{N}}$ is an orthonormal system. The invertibility of the matrix $\Psi$
implies that we can transform $(e_{j})_{1\leq j\leq D_{N}}$ into an
orthonormal system with respect to $\langle\cdot,\cdot\rangle_{\star}$.

######  Lemma 5.1.

The collection $\bar{e}=(\bar{e}_{1},\dots,\bar{e}_{D_{N}})^{\top}$ given by

(36) $\bar{e}=\Psi^{-1/2}e,$

where $e=(e_{1},\dots,e_{D_{N}})^{\top}$, is an orthonormal system with
respect to $\langle\cdot,\cdot\rangle_{\star}$.

###### Proof.

Denoting the Kronecker delta as $\delta_{ij}$, we have

(37)
$\left<\bar{e}_{i},\bar{e}_{j}\right>_{\star}=\left<(\Psi^{-1/2}e)_{i},(\Psi^{-1/2}e)_{j}\right>_{\star}=\left(\Psi^{-1/2}\Psi\Psi^{-1/2}\right)_{ij}=\delta_{ij}.$

∎

####  5.3.2. Matrix concentration inequality for independent observations

First of all, we recall a result from [36, Corollary 6.1], which provides a
concentration inequality for operator norms of self-adjoint matrices.

######  Theorem 5.2.

([36, Corollary 6.1]) Let $(X_{i})_{1\leq i\leq N}$ be a sequence of
independent random variables taking values in a Polish space $\mathcal{X}$,
and let $H$ be a function from $\mathcal{X}^{N}$ into the space of $m\times m$
real self-adjoint matrices $\mathbb{H}^{m}$. Assume that there exists a
sequence $(A_{i})_{1\leq i\leq N}\subset\mathbb{H}^{m}$ that satisfies

(38)
$A_{i}^{2}\succcurlyeq\left(H(x_{1},\ldots,x_{l-1},x_{l},x_{l+1}\ldots,x_{N})-H(x_{1},\ldots,x_{l-1},x^{\prime}_{l},x_{l+1}\ldots,x_{N})\right)^{2}$

for each index $1\leq i\leq N$, where $x_{l}$ and $x^{\prime}_{l}$ range over
all possible values of $X_{l}$. Define
$\nu^{2}=\left\|\sum_{i=1}^{N}A_{i}^{2}\right\|_{\text{\rm op}}$. Then it
holds that

(39)
$\mathbb{P}\left(\left\|H(X_{1},\ldots,X_{N})-\mathbb{E}{H(X_{1},\ldots,X_{N})}\right\|_{\text{\rm
op}}>t\right)\leq m\exp\left(-\frac{t}{2\nu^{2}}\right).$

We begin by considering the random matrix $\overline{\Psi}_{N}$ which has been
constructed from the i.i.d. observations $\overline{X}^{i,N}$, $i=1,\ldots,N$,
drawn from the mean field limit Eq. 3 instead of mutually dependent particles
(thus, we work under $\overline{\mathbb{P}}^{N}$). That is,

(40)
$(\overline{\Psi}_{N})_{jk}=\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}e_{j}\star\mu_{t}^{N}(\overline{X}_{t}^{i,N})e_{k}\star\mu_{t}^{N}(\overline{X}_{t}^{i,N})dt$

for $1\leq j,k\leq D_{N}$. As a slight abuse of notation, we denote the
corresponding bilinear form as $\left<\cdot,\cdot\right>_{N}$. In what
follows, we prove that 3.1 holds for $\overline{\Psi}_{N}$ and then transition
to the particle system in the next section.

###### Proof of 3.1.

(The independent case) (i) Here we would like to prove the inequality

(41)
$\displaystyle\mathbb{P}\left(\left\|\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}-\Psi^{-1/2}\mathbb{E}{\overline{\Psi}_{N}}\Psi^{-1/2}\right\|_{\operatorname{op}}\geq\frac{1}{2}\right)$
$\displaystyle\leq D_{N}\exp\left(-\frac{c_{\eta,T}\eta
NT}{4L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}}\right).$

via 5.2. Recall that by 5.1

$\left(\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}\right)_{jk}=\left<\bar{e}_{j},\bar{e}_{k}\right>_{N},$

where $\bar{e}=\Psi^{-1/2}e$. We consider the space
$\mathcal{X}=C([0,T],\mathbb{R})$ and the map $H$ given by

$\displaystyle H\colon\left(C([0,T],\mathbb{R})\right)^{N}$
$\displaystyle\to\mathbb{H}^{D_{N}},$
$\displaystyle\left(x_{1},\dots,x_{N}\right)$
$\displaystyle\mapsto\left(\frac{1}{NT}\sum_{i=1}^{N}\int_{0}^{T}\bar{e}_{j}\star\delta_{x(t)}(x_{i}(t))\bar{e}_{k}\star\delta_{x(t)}(x_{i}(t))\textnormal{d}t\right)_{1\leq
j,k\leq D_{N}}$

where $\delta_{x(t)}=\frac{1}{N}\sum_{j=1}^{N}\delta_{x_{j}(t)}$. We have that

$\displaystyle\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}-\Psi^{-1/2}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[\overline{\Psi}_{N}\right]\Psi^{-1/2}$
$\displaystyle=H\left(\overline{X}^{1,N},\ldots,\overline{X}^{N,N}\right)-\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[H\left(\overline{X}^{1,N},\ldots,\overline{X}^{N,N}\right)\right].$

For $x=\left(x_{1},\dots,x_{N}\right)\in\left(C([0,T],\mathbb{R})\right)^{N}$
we denote by $\tilde{x}$ the same vector where the $l$-th entry is replaced by
${x_{l}}^{\prime}$. We need to study $\Delta
H_{jk}:=\left(H(x)-H(\tilde{x})\right)_{jk}$, which is given by

$\displaystyle\Delta H_{jk}$
$\displaystyle=\frac{1}{TN^{3}}\sum_{i,r_{1},r_{2}=1}^{N}\int_{0}^{T}\bar{e}_{j}(x_{i}(t)-x_{r_{1}}(t))\bar{e}_{k}(x_{i}(t)-x_{r_{2}}(t))$
$\displaystyle-\bar{e}_{j}(\tilde{x}_{i}(t)-\tilde{x}_{r_{1}}(t))\bar{e}_{k}(\tilde{x}_{i}(t)-\tilde{x}_{r_{2}}(t))dt.$

If either none or all of the indices are equal to $l$, the difference
vanishes. Therefore, there are at most $3N(N-1)$ non-zero terms. Consequently,
for any $1\leq l\leq N$, we can bound the difference as

$|\Delta
H_{jk}|\leq\frac{6}{N}\left\|\bar{e}_{j}\right\|_{\infty}\left\|\bar{e}_{k}\right\|_{\infty}.$

Moving on to the spectral norm of $\Delta H$, we have for any
$y\in\mathbb{R}^{D_{N}}$:

$\displaystyle y^{\top}\Delta H^{2}y$
$\displaystyle=\sum_{j,k=1}^{D_{N}}y_{j}y_{k}\sum_{l=1}^{D_{N}}\Delta
H_{jl}\Delta
H_{kl}\leq\frac{36}{N^{2}}\sum_{j,k=1}^{D_{N}}|y_{j}y_{k}|\sum_{l=1}^{D_{N}}\left\|\bar{e}_{j}\right\|_{\infty}\left\|\bar{e}_{k}\right\|_{\infty}\left\|\bar{e}_{l}\right\|_{\infty}^{2}$
$\displaystyle\leq\frac{36}{N^{2}}\left(\sum_{j=1}^{D_{N}}\left\|\bar{e}_{j}\right\|_{\infty}^{2}\right)^{2}\left\|y\right\|^{2}\leq\frac{36}{N^{2}}L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}\left\|y\right\|^{2}.$

This means

$A_{l}^{2}:=\frac{36}{N^{2}}L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}I_{D_{N}}$

satisfies the condition $A_{l}^{2}\succcurlyeq\Delta H^{2}$ for all $1\leq
l\leq N$. We also define
$\nu^{2}:=\frac{36}{N}L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}$.
Now we apply the concentration inequality of 5.2 and deduce the statement Eq.
41, which completes this part of the proof.

(ii) To finish the proof of 3.1(i) in the independent case we observe that

$\displaystyle\left\|\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}-I_{D_{N}}\right\|_{\operatorname{op}}$
$\displaystyle\leq\left\|\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}-\Psi^{-1/2}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[\overline{\Psi}_{N}\right]\Psi^{-1/2}\right\|_{\operatorname{op}}$
$\displaystyle+\left\|\Psi^{-1/2}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[\overline{\Psi}_{N}\right]\Psi^{-1/2}-I_{D_{N}}\right\|_{\operatorname{op}}.$

Thus we only need to show the convergence

(42)
$\left\|\Psi^{-1/2}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[\overline{\Psi}_{N}\right]\Psi^{-1/2}-I_{D_{N}}\right\|_{\operatorname{op}}\to
0\qquad\text{as }N\to\infty.$

Since the particles are exchangeable and independent, we have

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[({\overline{\Psi}_{N}})_{jk}\right]$
$\displaystyle=\frac{1}{TN^{3}}\sum_{i,r_{1},r_{2}=1}^{N}\int_{0}^{T}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{j}(\overline{X}_{t}^{i,N}-\overline{X}_{t}^{r_{1},N})e_{k}(\overline{X}_{t}^{i,N}-\overline{X}_{t}^{r_{2},N})\right]dt$
$\displaystyle=\frac{1}{TN^{2}}\int_{0}^{T}\left((N-1)^{2}\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{j}\star\mu_{t}(\overline{X}^{1,N}_{t})e_{k}\star\mu_{t}(\overline{X}^{1,N}_{t})\right]+e_{j}(0)e_{k}(0)\right.$
$\displaystyle\quad+\left.(N-1)\left(\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{j}\star\mu_{t}(\overline{X}^{1,N}_{t})\right]e_{k}(0)+\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{k}\star\mu_{t}(\overline{X}^{1,N}_{t})\right]e_{j}(0)\right)\right)dt$
$\displaystyle=\frac{(N-1)^{2}}{N^{2}}\Psi_{jk}+\frac{e_{j}(0)e_{k}(0)}{N^{2}}$
$\displaystyle\quad+\frac{N-1}{TN^{2}}\int_{0}^{T}\left(\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{j}\star\mu_{t}(\overline{X}^{1,N}_{t})\right]e_{k}(0)+\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[e_{k}\star\mu_{t}(\overline{X}^{1,N}_{t})\right]e_{j}(0)\right)dt.$

Hence, we deduce that

(43)
$\left|\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[({\overline{\Psi}_{N}})_{jk}\right]-\Psi_{jk}\right|\lesssim\frac{1}{N}\left((I_{D_{N}})_{jk}+\left\|e_{j}\right\|_{\infty}\left\|e_{k}\right\|_{\infty}\right).$

Now, the spectral norm can be bounded through 4:

(44)
$\displaystyle\left\|I_{D_{N}}-\Psi^{-1/2}\mathbb{E}{\overline{\Psi}_{N}}\Psi^{-1/2}\right\|_{\operatorname{op}}$
$\displaystyle\lesssim\frac{1}{N}\left(1+\left\|\Psi^{-1}\right\|_{\operatorname{op}}\left(\sum_{j,k=1}^{D_{N}}\left\|e_{j}\right\|_{\infty}^{2}\left\|e_{k}\right\|_{\infty}^{2}\right)^{1/2}\right)$
$\displaystyle\lesssim\frac{L_{N}\left\|\Psi^{-1}\right\|_{\operatorname{op}}}{N}$
$\displaystyle\lesssim\frac{1}{\sqrt{N}\log(N)}.$

This gives the claim Eq. 42 and thus the validity of 3.1(i) in the independent
case.

(iii) On $\Lambda_{N}^{c}$, the following inequalities hold:

$\displaystyle c_{\eta,T}\frac{NT}{\log(NT)}$ $\displaystyle\leq
L_{N}^{2}\left(\left\|\Psi^{-1}\right\|_{\operatorname{op}}+\left\|\Psi^{-1}-\overline{\Psi}_{N}^{-1}\right\|_{\operatorname{op}}\right)^{2}$
$\displaystyle\leq
2L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}+2L_{N}^{2}\left\|\Psi^{-1}-\overline{\Psi}_{N}^{-1}\right\|_{\operatorname{op}}^{2}.$

So, if 4 holds, we deduce that

$\left\|\Psi^{-1}-\overline{\Psi}_{N}^{-1}\right\|_{\operatorname{op}}^{2}\geq\frac{c_{\eta,T}}{4L_{N}^{2}}\frac{NT}{\log(NT)}\geq\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2},$

and [14, Proposition 4 (ii)] yields the claim. ∎

####  5.3.3. From independent observations to the particle system

To transfer the results of 3.1 from the independent case to the particle
system introduced in Eq. 1, we use an elegant transformation of measure
argument investigated in [17]. We recall the probability measures
$\mathbb{P}^{N}$ and $\overline{\mathbb{P}}^{N}$ that have been introduced in
Section 5.1. According to [17, Theorem 18, eq. (35)], there exists a
$k\in\mathbb{N}$ such that

(45) $\mathbb{P}^{N}(A)\leq
C_{\varphi}^{k(k+1)/8}\overline{\mathbb{P}}^{N}(A)^{k/4}\quad\forall
A\in\mathcal{F}_{T},$

where $C_{\varphi}$ is a constant independent of $N$ and bounded by

(46) $C_{\varphi}\leq 1+\sum_{p=1}^{\infty}(\delta
a_{\varphi})^{p}\left\|\varphi\right\|_{\text{Lip}}^{2p},$

where $a_{\varphi}$ depends exponentially on
$\left\|\varphi\right\|_{\text{Lip}}$ and $\delta$ needs to be chosen.
Clearly,

$\delta<\left(2a_{\varphi}\left\|\varphi\right\|_{\text{Lip}}^{2}\right)^{-1}$

gives us an upper bound for $C_{\varphi}$, which is uniform over the class
$\Phi$. Therefore, we have

$\displaystyle\mathbb{P}^{N}\left(\left\|\Psi^{-1/2}\Psi_{N}\Psi^{-1/2}-I_{D_{N}}\right\|_{\operatorname{op}}>\frac{1}{2}\right)$
$\displaystyle\lesssim\overline{\mathbb{P}}^{N}\left(\left\|\Psi^{-1/2}\overline{\Psi}_{N}\Psi^{-1/2}-I_{D_{N}}\right\|_{\operatorname{op}}>\frac{1}{2}\right)^{k/4}$
$\displaystyle\lesssim
D_{N}^{k/4}\exp\left(-\frac{kc_{\eta,T}}{16L_{N}^{2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}}\eta
NT\right),$

meaning that 3.1 is true for the particle system Eq. 1. This completes the
proof of Proposition 3.1.

###  5.4. Proof of Theorem 3.2

We begin by considering the $\mathbb{R}^{D_{N}}$-valued random vector
$E_{N}=Z_{N}-\left<e,\varphi\right>_{N}$, that is

$(E_{N})_{j}=\frac{\sigma}{NT}\sum_{i=1}^{N}\int_{0}^{T}e_{j}\star\mu_{t}^{N}(X_{t}^{i,N})dW_{t}^{i}.$

By the Itô isometry,

(47)
$\mathbb{E}\left[(E_{N})_{j}(E_{N})_{k}\right]=\frac{\sigma^{2}}{NT}\Psi_{jk}+O(N^{-2}).$

######  Lemma 5.3.

For $E_{N}$ defined as above, we have

$\mathbb{E}\left[\left\|E_{N}\right\|^{4}\right]\lesssim\frac{D_{N}L_{N}^{2}}{(NT)^{2}}.$

###### Proof.

Using Jensen’s inequality, the Burkholder-Davis-Gundy inequality, and
independence of the $W^{i}$, we get

$\displaystyle\mathbb{E}\left[\left\|E_{N}\right\|^{4}\right]$
$\displaystyle\leq\sigma^{4}D_{N}\sum_{j=1}^{D_{N}}\mathbb{E}\left[\left(E_{N}\right)_{j}^{4}\right]\lesssim\sigma^{4}D_{N}\sum_{j=1}^{D_{N}}\mathbb{E}\left[\left(\frac{1}{(NT)^{2}}\sum_{i=1}^{N}\int_{0}^{T}\left(e_{j}\star\mu_{t}^{N}(X_{t}^{i,N})\right)^{2}\textnormal{d}t\right)^{2}\right]$
$\displaystyle\leq\frac{\sigma^{4}D_{N}}{N^{2}T^{3}}\sum_{j=1}^{D_{N}}\sum_{i=1}^{N}\int_{0}^{T}\mathbb{E}\left[\left(e_{j}\star\mu_{t}^{N}(X_{t}^{i,N})\right)^{4}\right]\textnormal{d}t$
$\displaystyle\leq\frac{\sigma^{4}D_{N}}{(NT)^{2}}\sum_{j=1}^{D_{N}}\left\|e_{j}\right\|_{\infty}^{4}\lesssim\frac{D_{N}L_{N}^{2}}{(NT)^{2}},$

which is the claim. ∎

For the asymptotic behaviour in $\left\|\cdot\right\|_{\star}$ and
$\left\|\cdot\right\|_{N}$, we consider orthogonal projections and their error
with respect to our introduced norms. We begin with some intermediate
estimates.

######  Proposition 5.4.

Let $\Pi^{N}:L^{2}(\|\cdot\|_{\star})\to S_{N}$ (resp. $\Pi^{\star}$) be the
orthogonal projection with respect to $\left\|\cdot\right\|_{N}$ (resp.
$\left\|\cdot\right\|_{\star}$) and recall that
$\Pi^{\star}\varphi=\varphi_{\star}$. Suppose that 4 is satisfied. Then we
have

  1. (i)

$\mathbb{E}\left[\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}1_{\Lambda_{N}\cap\Omega_{N}}\right]\lesssim\frac{D_{N}}{NT}.$

  2. (ii)

$\mathbb{E}\left[\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]\lesssim\frac{1}{NT}.$

  3. (iii)

$\mathbb{E}\left[\left\|\Pi^{N}\left(\varphi-\varphi_{\star}\right)\right\|_{\star}^{2}1_{\Omega_{N}}\right]\lesssim
N^{-1/2}(1+L_{N})\|\varphi-\varphi_{\star}\|_{\star\star}^{2}+O(N^{-1})+o\left(\|\varphi-\varphi_{\star}\|_{\star}^{2}\right)$.

######  Remark 5.5.

Depending on the choice of $\eta$, (ii) actually decays faster than stated.
However, as (i) is the slowest term in 3.2, a convergence rate of $(NT)^{-1}$
is sufficient. We will see in our proofs that this requires $\eta\geq 3$, see
Section 5.4. Considering we are analysing the projection error in its
respective norm, the proof of this lemma hardly differs from that of [15,
Lemma 6.3]. ∎

###### Proof of 5.4.

(i) As $\Pi^{N}\varphi-\varphi=\operatorname*{argmin}_{f\in
S_{N}}\left\|f-\varphi\right\|_{N}$, a straightforward computation gives

(48)
$\Pi^{N}\varphi=\sum_{j=1}^{D_{N}}(\widehat{a}_{N})_{j}e_{j},\quad\widehat{a}_{N}=\Psi_{N}^{-1}\left<e,\varphi\right>_{N}.$

By definition of $E_{N}$, $\theta_{N}-\widehat{a}_{N}=\Psi_{N}^{-1}E_{N}$.
Hence,

(49)
$\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}=\left\|\sum_{j=1}^{D_{N}}\left(\Psi_{N}^{-1}E_{N}\right)_{j}e_{j}\right\|_{N}^{2}=E_{N}^{\top}\Psi_{N}^{-1}E_{N}.$

On $\Omega_{N}$, all eigenvalues of $\Psi^{-1/2}\Psi_{N}\Psi^{-1/2}$ are in
$[\frac{1}{2},\frac{3}{2}]$, which implies that

(50) $\left\|\Psi^{1/2}\Psi_{N}^{-1}\Psi^{1/2}\right\|_{\operatorname{op}}\leq
2,$

such that $E_{N}^{\top}\Psi_{N}^{-1}E_{N}1_{\Omega_{N}}\leq
2E_{N}^{\top}\Psi^{-1}E_{N}$. Together with Eq. 47 and 5.3, we have deduced by
4:

$\displaystyle\mathbb{E}\left[\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}1_{\Lambda_{N}\cap\Omega_{N}}\right]$
$\displaystyle\leq
2\mathbb{E}\left[E_{N}^{\top}\Psi^{-1}E_{N}\right]\lesssim\sum_{j,k=1}^{D_{N}}\left(\Psi^{-1}\right)_{jk}\left(\frac{1}{NT}\Psi_{jk}+\frac{1}{N^{2}}\right)$
$\displaystyle\lesssim\frac{D_{N}}{NT}+\frac{D_{N}\left\|\Psi^{-1}\right\|_{\operatorname{op}}}{N^{2}}\lesssim\frac{D_{N}}{NT}+\frac{D_{N}}{L_{N}\sqrt{N^{3}\log(NT)}}$
$\displaystyle\lesssim\frac{D_{N}}{NT}.$

(ii) By 3.1, Eq. 49 and 5.3, we have

$\displaystyle\mathbb{E}\left[\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]$
$\displaystyle\leq\mathbb{E}\left[\left\|\Psi_{N}^{-1}\right\|_{\operatorname{op}}^{2}1_{\Lambda_{N}}\left\|E_{N}\right\|^{4}\right]^{1/2}\mathbb{P}\left(\Omega_{N}^{c}\right)^{1/2}$
$\displaystyle\leq\sqrt{D_{N}}(NT)^{-(\eta/2+1)}.$

(iii) We define $g=\varphi-\varphi_{\star}$ and write the projection of
$\Pi^{N}g$ according to Eq. 48 as an inner product. Using 5.1, Parseval’s
identity, and Eq. 50 we deduce the following inequalities on $\Omega_{N}$.

$\displaystyle\left\|\Pi^{N}g\right\|_{\star}^{2}$
$\displaystyle=\left\|{e^{\top}}\Psi_{N}^{-1}\left<e,g\right>_{N}\right\|^{2}_{\star}1_{\Omega_{N}}=\left\|\bar{e}^{\top}\Psi^{1/2}\Psi_{N}^{-1}\Psi^{1/2}\right\|\left<\bar{e},g\right>_{\star}^{2}$
$\displaystyle=\left\|\Psi^{1/2}\Psi_{N}^{-1}\Psi^{1/2}\left<\bar{e},g\right>_{N}\right\|^{2}\leq
4\left\|\left<\bar{e},g\right>_{N}\right\|^{2}$ $\displaystyle\leq
4\left\|\Psi^{-1}\right\|_{\operatorname{op}}\sum_{j=1}^{D_{N}}\left<e_{j},g\right>_{N}^{2}.$

Due to the identity $g=\varphi-\varphi_{\star}$, we have that
$\left<e_{j},g\right>_{\star}=0$ for all $j=1,\ldots,D_{N}$. Hence,

(51)
$\displaystyle\sum_{j=1}^{D_{N}}\mathbb{E}\left[\left<e_{j},g\right>_{N}^{2}\right]=\sum_{j=1}^{D_{N}}\mathbb{E}\left[\left(\left<e_{j},g\right>_{N}-\left<e_{j},g\right>_{\star}\right)^{2}\right].$

For the latter term we want to use the decomposition Eq. 28. However, we note
that $\|g\|_{\infty}$ is not necessarily uniformly bounded in $N$, and thus we
need a precise estimate of this norm. We recall that
$\|\varphi\|_{\star}\leq\|\varphi\|_{\infty}\leq K_{1}$ for all
$\varphi\in\Phi$. Also note the identity

$\varphi_{\star}=\sum_{j=1}^{D_{N}}\left<\bar{e}_{j},\varphi\right>_{\star}\bar{e}_{j}.$

Consequently, we deduce that

$\displaystyle\left\|\varphi_{\star}\right\|_{\infty}$
$\displaystyle\leq\sum_{j=1}^{D_{N}}|\left<\bar{e}_{j},\varphi\right>_{\star}|\cdot\|\bar{e}_{j}\|_{\infty}\leq\left(\sum_{j=1}^{D_{N}}\left<\bar{e}_{j},\varphi\right>_{\star}^{2}\right)^{1/2}\left(\sum_{j=1}^{D_{N}}\|\bar{e}_{j}\|_{\infty}^{2}\right)^{1/2}$
$\displaystyle\leq
L_{N}^{1/2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{1/2}\left(\sum_{j=1}^{D_{N}}\left<\bar{e}_{j},\varphi\right>_{\star}^{2}\right)^{1/2}.$

On the other hand, we have

$\left\|\varphi_{\star}\right\|_{\star}^{2}=\sum_{j=1}^{D_{N}}\left<\bar{e}_{j},\varphi\right>_{\star}^{2}\leq\left\|\varphi\right\|_{\star}^{2}\leq
K_{1}^{2}.$

Hence, we obtain the inequality

(52)
$\displaystyle\left\|g\right\|_{\infty}\lesssim\left(1+L_{N}^{1/2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{1/2}\right).$

In the next step, we use the decomposition Eq. 28 and the polarisation
identity
$\left<f,g\right>_{\star}=(\|f+g\|_{\star}^{2}-\|f-g\|_{\star}^{2})/4$, to get

(53)
$\displaystyle\left<e_{j},g\right>_{N}-\left<e_{j},g\right>_{\star}=U_{1,N}(g,e_{j})+U_{2,N}(g,e_{j})+U_{3,N}(g,e_{j})+O\left(N^{-1}\|g\|_{\infty}\|e_{j}\|_{\infty}\right),$

where $U_{k,N}(g,e_{j}):=(U_{k,N}(g+e_{j})-U_{k,N}(g-e_{j}))/4$ for $k=1,2,3$.
Now, we need to estimate the variance of the terms $U_{k,N}(g,e_{j})$. For
this purpose we will use the same change of measure device as proposed in
Section 5.1. In particular, due to Cauchy-Schwarz inequality Eq. 33, it
suffices to find an appropriate bound for the fourth moment of
$U_{k,N}(g,e_{j})$, $k=1,2,3$, under the probability measure
$\overline{\mathbb{P}}^{N}$.

We start with the U-statistics $U_{2,N}(g,e_{j})$ and $U_{3,N}(g,e_{j})$.
Replacing $g$ (resp. $e_{j}$) by $g/\|g\|_{\infty}$ (resp. by
$e_{j}/\|e_{j}\|_{\infty}$), we conclude as in Section 5.1:

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[U_{2,N}(g,e_{j})^{4}\right]^{1/2}$
$\displaystyle\lesssim\|g\|_{\infty}^{2}\|e_{j}\|_{\infty}^{2}N^{-2},$ (54)
$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[U_{3,N}(g,e_{j})^{4}\right]^{1/2}$
$\displaystyle\lesssim\|g\|_{\infty}^{2}\|e_{j}\|_{\infty}^{2}N^{-3}.$

Next, we treat the term $U_{1,N}(g,e_{j})$. Under $\overline{\mathbb{P}}^{N}$
we deal with i.i.d. random variables. Observe that for a statistic
$W_{N}:=N^{-1}\sum_{i=1}^{N}Z_{i}$, where $Z_{i}$’s are i.i.d. random
variables with $\mathbb{E}[Z_{1}]=0$ and $\mathbb{E}[Z_{1}^{4}]<\infty$, we
have that
$\mathbb{E}[W_{N}^{4}]=3N^{-2}\mathbb{E}[Z_{1}^{2}]^{2}+N^{-3}\mathbb{E}[Z_{1}^{4}]$.
Using this observation and applying Hölder inequality, a straightforward
computation shows that

(55)
$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[U_{1,N}(g,e_{j})^{4}\right]^{1/2}\lesssim\|e_{j}\|_{\infty}^{2}\left(\|g\|_{\star}^{2}+\|g\|_{\star\star}^{2}\right)\left(N^{-1}+N^{-3/2}\|e_{j}\|_{\infty}^{2}\|g\|_{\infty}^{2}\right).$

Now, we put everything together. According to Eq. 33, we deduce that

$\mathbb{E}_{\mathbb{P}^{N}}\left[U_{k,N}(g,e_{j})^{2}\right]\leq
C\mathbb{E}_{\overline{\mathbb{P}}^{N}}\left[U_{1,N}(g,e_{j})^{4}\right]^{1/2}\qquad\text{for
}k=1,2,3.$

Plugging the inequalities Eq. 54 and Eq. 55 into Eq. 51, and taking into
account the bound in Eq. 52, we conclude that

$\displaystyle\mathbb{E}\left[\left\|\Pi^{N}\left(\varphi-\varphi_{\star}\right)\right\|_{\star}^{2}1_{\Omega_{N}}\right]$
$\displaystyle\lesssim\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}L_{N}^{2}\left(N^{-2}+N^{-3}\right)$
$\displaystyle+N^{-1}\left\|\Psi^{-1}\right\|_{\operatorname{op}}L_{N}\left(\|g\|_{\star}^{2}+\|g\|_{\star\star}^{2}\right)$
$\displaystyle+N^{-3/2}\left\|\Psi^{-1}\right\|_{\operatorname{op}}^{2}L_{N}^{3}\left(\|g\|_{\star}^{2}+\|g\|_{\star\star}^{2}\right).$

Applying Assumption 4, we obtain the statement of the proposition. ∎

Now, we proceed with the proof of Theorem 3.2. Let us start with the
asymptotic result in the empirical norm. We split

$\displaystyle\left\|\widehat{\varphi}_{N}-\varphi\right\|_{N}^{2}$
$\displaystyle=\inf_{f\in
S_{N}}\left\|f-\varphi\right\|_{N}^{2}+\left\|\widehat{\varphi}_{N}-\Pi^{N}\varphi\right\|_{N}^{2}$
$\displaystyle=\inf_{f\in
S_{N}}\left\|f-\varphi\right\|_{N}^{2}+\left\|\varphi_{N}-\Pi^{N}\varphi\right\|_{N}^{2}\left(1_{\Lambda_{N}\cap\Omega_{N}}+1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right)+\left\|\varphi\right\|_{N}^{2}1_{\Lambda_{N}^{c}}$
$\displaystyle=:\inf_{f\in
S_{N}}\left\|f-\varphi\right\|_{N}^{2}+\operatorname{I}_{1}+\operatorname{I}_{2}+\operatorname{I}_{3}.$

Recall that $\mathbb{E}[\operatorname{I}_{1}]$ and
$\mathbb{E}[\operatorname{I}_{2}]$ are treated in 5.4. Furthermore, we have
$\mathbb{E}{\operatorname{I}_{3}}\lesssim 1/N$ due to 3.1. This gives us

$\mathbb{E}\left[\left\|\widehat{\varphi}_{N}-\varphi\right\|_{N}^{2}\right]\lesssim\mathbb{E}\left[\inf_{f\in
S_{N}}\left\|f-\varphi\right\|_{N}^{2}\right]+\frac{D_{N}}{NT}.$

Taking the supremum over $\varphi\in\Phi$ gives the first claim.

For the asymptotic results in $\left\|\cdot\right\|_{\star}$, we split

$\displaystyle\left\|\widehat{\varphi}_{N}-\varphi\right\|_{\star}^{2}$
$\displaystyle=\left\|\widehat{\varphi}_{N}-\varphi\right\|_{\star}^{2}\left(1_{\Lambda_{N}\cap\Omega_{N}}+1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right)+\left\|\varphi\right\|_{\star}^{2}1_{\Lambda_{N}^{c}}$
$\displaystyle=:\operatorname{II}_{1}+\operatorname{II}_{2}+\operatorname{II}_{3}.$

For $\operatorname{II}_{1}$, we set $g=\varphi-\varphi_{\star}$. Note that
$\Pi^{N}\Pi^{\star}=\Pi^{\star}$, therefore

$\Pi^{N}\varphi-\varphi=\Pi^{N}(g+\Pi^{\star}\varphi)-\varphi=\Pi^{N}g-g.$

So we split the terms in $\operatorname{II}_{1}$ into

$\displaystyle\left\|\widehat{\varphi}_{N}-\varphi\right\|_{\star}^{2}$
$\displaystyle\leq\left\|\widehat{\varphi}_{N}-\Pi^{N}\varphi\right\|_{\star}^{2}+\left\|\Pi^{N}\varphi-\varphi\right\|_{\star}^{2}$
$\displaystyle\leq\left\|\widehat{\varphi}_{N}-\Pi^{N}\varphi\right\|_{\star}^{2}+\left\|\Pi^{N}g\right\|_{\star}^{2}+\left\|g\right\|_{\star}^{2}.$

Since $\left\|\cdot\right\|_{N}$ and $\left\|\cdot\right\|_{\star}$ are
equivalent on $\Omega_{N}$, we have

$\mathbb{E}\left[\left\|\widehat{\varphi}_{N}-\Pi^{N}\varphi\right\|_{\star}^{2}1_{\Lambda_{N}\cap\Omega_{N}}\right]\leq
2\mathbb{E}\left[\left\|\widehat{\varphi}_{N}-\Pi^{N}\varphi\right\|_{N}^{2}1_{\Lambda_{N}\cap\Omega_{N}}\right]\lesssim\frac{D_{N}}{NT}.$

By 5.4 (iii), we have

$\mathbb{E}\left[\left\|\Pi^{N}g\right\|_{\star}^{2}1_{\Lambda_{N}\cap\Omega_{N}}\right]+\left\|g\right\|_{\star}^{2}\lesssim(1+o(1))\left\|g\right\|_{\star}^{2}+N^{-1/2}(1+L_{N})\left\|g\right\|_{\star\star}^{2}+O(N^{-1}).$

Moving on to $\operatorname{II}_{2}$, we have

$\displaystyle\mathbb{E}\left[\left\|\widehat{\varphi}_{N}-\varphi\right\|_{\star}^{2}1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]\leq\mathbb{E}\left[\left\|\varphi_{N}\right\|_{\star}^{2}1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]+\left\|\varphi\right\|_{\star}^{2}\mathbb{P}\left(\Omega_{N}^{c}\right).$

Recalling the definition of $\varphi_{N}$ given in Eq. 10 and that
$(Z_{N})_{j}=(E_{N})_{j}+\langle e_{j},\varphi\rangle_{N}$, we have

$\displaystyle\left\|\varphi_{N}\right\|^{2}_{\star}$
$\displaystyle=\left(\Psi_{N}^{-1}Z_{N}\right)^{\top}\Psi\Psi_{N}^{-1}Z_{N}\leq\left\|\Psi_{N}^{-1}\right\|_{\operatorname{op}}^{2}\left\|\Psi\right\|_{\operatorname{op}}\left\|Z_{N}\right\|^{2}$
$\displaystyle=\left\|\Psi_{N}^{-1}\right\|_{\operatorname{op}}^{2}\left\|\Psi\right\|_{\operatorname{op}}\left(\left\|\left<e,\varphi\right>_{N}\right\|^{2}+\left\|E_{N}\right\|^{2}\right).$

Since $\varphi$ is bounded,

$\displaystyle\sum_{j=1}^{D_{N}}\mathbb{E}\left[\left<e_{j},\varphi\right>_{N}^{4}\right]$
$\displaystyle\leq\frac{1}{(NT)^{2}}\sum_{j=1}^{D_{N}}\int_{0}^{T}\mathbb{E}\left[e_{j}\star\mu^{N}_{t}(X^{1,N}_{t})^{4}\varphi\star\mu^{N}_{t}(X^{1,N}_{t})^{4}\right]\textnormal{d}t$
$\displaystyle\leq\sum_{j=1}^{D_{N}}\frac{\left\|e_{j}\right\|^{4}_{\infty}}{(NT)^{2}}\int_{0}^{T}\mathbb{E}\left[\varphi\star\mu^{N}_{t}(X^{1,N}_{t})^{4}\right]\textnormal{d}t\leq\frac{L_{N}^{2}\left\|\varphi\right\|_{\infty}^{4}}{(NT)^{2}},$

such that

$\displaystyle\mathbb{E}\left[\left(\left\|\left<e,\varphi\right>_{N}\right\|^{2}+\left\|E_{N}\right\|^{2}\right)1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[\left\|\left<e,\varphi\right>_{N}\right\|^{4}\right]^{1/2}+\mathbb{E}\left[\left\|E_{N}\right\|^{4}\right]^{1/2}\right)\mathbb{P}\left(\Omega_{N}^{c}\right)^{1/2}$
$\displaystyle\leq\left(\frac{\left\|\varphi\right\|_{\infty}^{2}L_{N}}{NT}+\frac{\sqrt{D}_{N}L_{N}}{NT}\right)(NT)^{-\frac{\eta-1}{2}}$
$\displaystyle\leq\left(\frac{\left\|\varphi\right\|_{\infty}^{2}}{\sqrt{D_{N}}}+1\right)L_{N}(NT)^{-\frac{\eta}{2}}.$

Finally, with 4 and 5.3, and the fact that
$\left\|\Psi\right\|_{\operatorname{op}}\leq L_{N}$, we obtain

$\displaystyle\mathbb{E}\left[\left\|\varphi_{N}\right\|_{\star}^{2}1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]$
$\displaystyle\leq\mathbb{E}\left[\left\|\Psi_{N}^{-1}\right\|_{\operatorname{op}}^{2}\left\|\Psi\right\|_{\operatorname{op}}\left(\left\|\left<e,\varphi\right>_{N}\right\|^{2}+\left\|E_{N}\right\|^{2}\right)1_{\Lambda_{N}\cap\Omega_{N}^{c}}\right]$
$\displaystyle\leq
c_{\eta,T}\left(\frac{\left\|\varphi\right\|_{\infty}^{2}}{\sqrt{D}_{N}}+1\right)(NT)^{-(\eta-1)/2}.$

Last but not least, we have that

$\mathbb{E}[\operatorname{II}_{3}]\lesssim\mathbb{P}(\Lambda_{N}^{c})$

which completes the proof due to Proposition 3.1.

###  5.5. Proof of 3.4

Let us denote by $\lambda_{\min}(Q)$ the smallest eigenvalue of a symmetric
matrix $Q\in\mathbb{R}^{D_{N}\times D_{N}}$. Since

$\displaystyle\lambda_{\min}(Q_{1}Q_{2})$
$\displaystyle\geq\lambda_{\min}(Q_{1})\lambda_{\min}(Q_{2})\qquad\text{and}$
$\displaystyle\lambda_{\min}\left(T^{-1}\int_{0}^{T}Q_{t}dt\right)$
$\displaystyle\geq T^{-1}\int_{0}^{T}\lambda_{\min}(Q_{t})dt,$

for all positive semi-definite matrices $Q_{1},Q_{2},(Q_{t})_{t\geq 0}$, we
obtain via Eq. 17:

$\displaystyle\lambda_{\min}(\Psi)$
$\displaystyle\geq\frac{1}{2A_{N}T}\int_{0}^{T}\min_{k=1,\ldots,D_{N}}(\widehat{\mu}_{t}(k/A_{N}))^{2}\lambda_{\min}(R_{t})dt$
$\displaystyle\geq\frac{\exp\left(-2D_{N}^{2}/(c_{2}A_{N}^{2})\right)}{2A_{N}T}\int_{0}^{T}g_{2}^{2}(t)\lambda_{\min}(R_{t})dt$

where $R_{t}\in\mathbb{R}^{D_{N}\times D_{N}}$ is a Toeplitz matrix given as
$R_{t}^{kl}=\widehat{\mu}_{t}((l-k)/A_{N})$. To deduce a lower bound for
$\lambda_{\min}(R_{t})$, we introduce the discrete-time Fourier transform

$F(\omega):=\sum_{n\in\mathbb{Z}}\widehat{\mu}_{t}(n/A_{N})\exp(-i\omega
n),\qquad\omega\in[-\pi,\pi].$

A well-known result states that

$\lambda_{\min}(R_{t})\geq\min_{\omega\in[-\pi,\pi]}F(\omega),$

and hence we need to compute $F(\omega)$. For this purpose, we will apply the
Poisson summation formula. Define the function
$f(x):=\widehat{\mu}_{t}(x/A_{N})\exp(-i\omega x)$ for $x\in\mathbb{R}$ and
$\mathcal{F}f(z):=\int_{\mathbb{R}}f(x)\exp(-i2\pi zx)dx$. The Poisson
summation formula states that

$F(\omega)=\sum_{n\in\mathbb{Z}}f(n)=\sum_{n\in\mathbb{Z}}\mathcal{F}f(n).$

By the Fourier inversion formula we obtain the identity

$\mathcal{F}f(n)=\int_{\mathbb{R}}\widehat{\mu}_{t}(x/A_{N})\exp\left(-ix(\omega+2\pi
n)\right)dx=\frac{A_{N}}{\sqrt{2\pi}}\mu_{t}\left(A_{N}(\omega+2\pi
n)/\pi\right).$

Due to Assumption 5, we deduce that

$\displaystyle F(\omega)$
$\displaystyle\geq\frac{A_{N}g_{1}(t)}{\sqrt{2\pi}}\sum_{n\in\mathbb{Z}}\exp\left(-\frac{A_{N}^{2}(\omega+2\pi
n)^{2}}{c_{1}\pi^{2}}\right)$
$\displaystyle\geq\frac{A_{N}g_{1}(t)}{\sqrt{2\pi}}\exp\left(-\frac{A_{N}^{2}}{c_{1}}\right)(1+o(1)).$

Putting things together, we conclude that

$\lambda_{\min}(\Psi)\geq\frac{\exp\left(-2D_{N}^{2}/(c_{2}A_{N}^{2})\right)\exp\left(-A_{N}^{2}/c_{1}\right)}{2\sqrt{2\pi}T}\int_{0}^{T}g_{1}(t)g_{2}(t)^{2}dt.$

This implies the statement of Proposition 3.4.

###  5.6. Proof of Proposition 3.5

Consider the decomposition

$\varphi=\sum_{j=1}^{\infty}c_{j}e_{j}=\sum_{j=1}^{D_{N}}c_{j}e_{j}+\sum_{j=D_{N}+1}^{\infty}c_{j}e_{j}=:\varphi_{1}+\varphi_{2}.$

In this scenario it holds that

$\varphi-\Pi^{\star}\varphi=\varphi_{2}-\Pi^{\star}\varphi_{2}.$

It trivially holds that

$\|\varphi_{2}\|_{\star\star}^{2}\leq\|\varphi_{2}\|_{\infty}^{2}\lesssim
A_{N}^{-1}\left(\sum_{j=D_{N}+1}^{\infty}|c_{j}|\right)^{2}.$

To estimate the term $\|\Pi^{\star}\varphi_{2}\|_{\star\star}^{2}$, we
introduce the decomposition

$\Pi^{\star}\varphi_{2}=\sum_{j=1}^{D_{N}}\langle\bar{e}_{j},\varphi_{2}\rangle_{\star}\bar{e}_{j}.$

Now, we deduce that

$\|\Pi^{\star}\varphi_{2}\|_{\star\star}^{2}\lesssim\|\Pi^{\star}\varphi_{2}\|_{\infty}^{2}.$

We obtain the inequality

$\|\Pi^{\star}\varphi_{2}\|_{\infty}\leq\sum_{j=1}^{D_{N}}|\langle\bar{e}_{j},\varphi_{2}\rangle_{\star}|\cdot\|\bar{e}_{j}\|_{\infty}$

In the next step we will find a bound for the term
$|\langle\bar{e}_{j},\varphi_{2}\rangle_{\star}|$. First, we note

$\langle\bar{e}_{j},\varphi_{2}\rangle_{\star}=\sum_{k=1}^{D_{N}}\sum_{l=D_{N}+1}\Psi^{-1/2}_{jk}c_{l}\left<e_{k},e_{l}\right>_{\star}.$

We already know that

$\left<e_{k},e_{l}\right>_{\star}=\frac{1}{2A_{N}T}\int_{0}^{T}\widehat{\mu}_{t}(-k/A_{N})\widehat{\mu}_{t}(l/A_{N})\widehat{\mu}_{t}((l-k)/A_{N})dt$

Due to Assumption 2, the leading order of the above term is achieved for
$l=D_{n}+1$ and $k=D_{n}/2$. Hence, we conclude that

$|\langle\bar{e}_{j},\varphi_{2}\rangle_{\star}|\lesssim
D_{N}\|\Psi^{-1}\|^{1/2}_{\text{op}}\exp\left(-\frac{3D_{N}^{2}\zeta^{2}}{4A_{N}^{2}}\right)\sum_{j=D_{N}+1}^{\infty}|c_{j}|$

Consequently, we obtain the inequality

$\|\Pi^{\star}\varphi_{2}\|_{\infty}\lesssim
A_{N}^{-1/2}D_{N}^{2}\|\Psi^{-1}\|_{\text{op}}\exp\left(-\frac{3D_{N}^{2}\zeta^{2}}{4A_{N}^{2}}\right)\sum_{j=D_{N}+1}^{\infty}|c_{j}|,$

which completes the proof.

##  Appendix A Maximal inequalities for $U$-statistics

Let $\mathcal{G}$ be a class of real-valued functions on $\mathbb{R}^{K}$ and
$(U_{N}(g):$ $g\in\mathcal{G})$ be $U$-process of the form

$\displaystyle
U_{N}(g)=\frac{1}{(N)_{K}}\sum_{i_{1},\ldots,i_{K}=1}^{N}g(X_{i_{1}},\ldots,X_{i_{K}})$

where $X_{1},\ldots,X_{N}$ is a sequence of iid random variables with common
distribution $\mathbb{P}$ and $(N)_{K}=N(N-1)\cdot\ldots\cdot(N-K+1).$ If for
each $g\in\mathcal{G},$

$\mathbb{E}_{X\sim\mathbb{P}}[g(x_{1},\ldots,x_{i-1},X,x_{i+1},\ldots,x_{K})]=0,\quad
i=1,\ldots,K,$

for arbitrary $x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{K}\in\mathbb{R},$ then
the process $(U_{N}(g),$ $g\in\mathcal{G})$ is called degenerate.

######  Theorem A.1.

Suppose that $\mathcal{G}$ is a class of uniformly bounded functions with
$\sup_{g\in\mathcal{G}}\|g\|_{\infty}\leq G$ and the $U$-process $(U_{N}(g),$
$g\in\mathcal{G})$ is degenerate. Then it holds

$\displaystyle\Bigl{\\{}\mathbb{E}\sup_{g\in\mathcal{G}}\left|N^{K/2}U_{N}(g)\right|^{q}\Bigr{\\}}^{1/q}$
$\displaystyle\lesssim
Gp^{K}\,\mathsf{DI}(\mathcal{G},\|\cdot\|_{\infty},\psi_{p/K})$

for any $p\geq q.$

###### Proof.

We can apply the maximal inequality from [39, Section 5] and note that for
uniformly bounded classes $\mathcal{G}$

$d_{U_{N}}:=\bigl{[}U_{N}\bigl{(}|f-g|^{2}\bigr{)}\bigr{]}^{1/2}\leq\|f-g\|_{\infty}$

as well as $U_{N}\bigl{(}|g|^{2}\bigr{)}\leq G^{2}$ with probability $1$ for
all $f,g\in\mathcal{G}.$ ∎

### Acknowledgement

Mark Podolskij and Shi-Yuan Zhou gratefully acknowledge financial support of
ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High
Dimensional Diffusions”.

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