\section{Introduction} \label{sec1}
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In the past fifty years  diffusion processes found numerous applications in natural and social sciences, and a variety of statistical methods have been investigated in the setting of SDEs during the last few decades. Maximum likelihood estimation and Bayesian approach are the most well established parametric methods in the literature; we refer to the monograph \cite{kutoyants2013statistical}. 
When the likelihood function is not available in a closed form, quasi likelihood methods provide an alternative approach to parameter estimation, see \cite{chang2011approximate} and references therein. The most recent contributions to nonparametric inference for diffusions are \cite{nickl2017nonparametric,strauch2018adaptive}.
These belong to the most successful tools when analysing
estimation problems in different observation schemes of a diffusion.

Many diffusion models in natural and applied sciences can be viewed as continuous-time processes with complex and nonlinear probabilistic structure. For example, in statistical mechanics nonlinear diffusion models and particle systems have long and successful history. In general, nonlinear Markov processes are stochastic processes whose transition functions may depend not only on the current state of the process but also on the
current distribution of the process. These processes were introduced by
McKean~\cite{mckean1966class} to model plasma dynamics. Later nonlinear Markov
processes were studied by a number of authors; we mention here the books of
Kolokoltsov~\cite{kolokoltsov2010nonlinear} and
Sznitman~\cite{sznitman1991topics}. These processes arise naturally in the
study of the limit behavior of a large number of weakly interacting particles and have a wide range of applications, including financial
mathematics, population dynamics, and neuroscience (see, e.g.,
\cite{frank2005nonlinear} and the references therein). In this context the mean field theory has been employed to bridge the interaction of particles at the microscopic scale and the mesoscopic features of the system. From a probabilistic point of view propagation of chaos, fluctuation analysis, and large deviations have been investigated for a variety of mean field models and nonlinear diffusions. 
\par
In recent years, one witnessed a growing interest in statistical problems for high dimensional diffusions
in general and McKean-Vlasov (MV) SDEs in particular. Statistical inference for high dimensional Ornstein-Uhlenbeck models have been investigated in \cite{CMP20,GM19}. 
A parametric problem of estimating the coefficients of a MV-SDE under a small noise assumption have been  studied in \cite{GL20,ren2019least}.
Our current work is mostly related  to  a recent paper  \cite{dellamaestra2021nonparametric}, where  based on observation of a trajectory of an interacting particle system over a fixed time horizon,  the authors study nonparametric estimation of the solution (density) of the associated nonlinear Fokker-Planck equation, together with the drift function. The underlying statistical problem turns out to be rather challenging and   \cite{dellamaestra2021nonparametric} contains only partial results. In particular, the problem of estimating  a distribution dependent drift function of a MV-SDE from the observations of the corresponding particle system at time \(T>0\) has not been yet studied in the  literature. 
\par
In this paper we  consider the case where the drift  has a semiparametric form consisting of a polynomial part, a trigonometric part and a nonparametric interaction function convolved with an unknown marginal distribution of  the underlying MV-SDE.  The goal of this research is twofold: first to propose a kernel type estimator for the drift function based on the empirical characteristic function of the particles; and, second, to  study its properties. We derive upper bounds on \(L^2\) risk of the proposed estimator and show that these bounds are essentially optimal in minimax sense for a properly chosen functional class of drift functions. In particular, we show that the convergence rates of our estimator are logarithmic under a polynomial tail behaviour of the non-parametric part of the interaction function. Our approach is based on a rigorous analysis of the related inverse problem for the underlying stationary Fokker-Planck equation and makes use of the probabilistic properties of the model obtained in \cite{cattiaux2008probabilistic,malrieu2003convergence}. To the best of our knowledge, this is the first work containing minimax optimal procedure  for semiparametric estimation  of the coefficients of MV-SDEs from discrete observations of the corresponding particle system and hence  fills an important gap in the current literature on  statistical inference for MV-SDEs. 
\par
The structure of the paper is as follows. In Section~\ref{sec2} we introduce the main setup and  recall some basic facts about  MV-SDEs and related particle systems. In Section~\ref{sec3} we formulate our main statistical problem and describe the estimation procedure. Section~\ref{sec4} is devoted to the convergence analysis of the proposed algorithm.  In particular, we derive upper bounds on \(L^2\) risk of the suggested drift estimator.  In Section~\ref{sec5} we complete our theoretical analysis  by providing   lower bounds for the nonparametric part of the model that essentially match upper bounds obtained  in Section~\ref{sec4}. 
Conclusions and outlook are presented in Section~\ref{secOut}. All proofs are collected in Section~\ref{sec6}.


\section{The particle system model and propagation of chaos} \label{sec2}
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Throughout the paper we consider a filtered probability space $(\Omega ,\mathcal{F},(\f_t)_{t\geq0},\P)$, on which all stochastic processes are defined. We focus on an $N$-dimensional system of stochastic differential equations given by
\begin{equation} \label{eq:sde-part}
X_{t}^{i,N}=X_{0}^{i}+B_{t}^{i}-\frac{1}{2N}\sum_{j=1}^{N}\int_{0}^{t}\varphi' (X_{s}^{i,N}-X_{s}^{j,N} )\,ds, \qquad 1 \le i \le N, \qquad t\geq0,
\end{equation}
where $B^{i} = (B^{i}_t)_{t \ge 0}$, $1 \le i \le N$, are independent
standard Brownian motions and $X_{0}^{i}$, $1 \le i \le N$, are i.i.d.\ random variables with distribution $\mu_0(dx)$.
Here the \textit{interaction function} $\varphi'$ denotes the derivative of the function $\varphi \in C^2(\mathbb{R})$, which satisfies the following assumption: 
\vspace{0.3 cm}
\begin{itemize}
\item[(A)] The function $\varphi$ is even (i.e. $\varphi(x)=\varphi(-x)$ for all $x\in \mathbb{R}$), strictly convex: 
\begin{equation} \label{convex}
\varphi''(x) \geq \lambda>0, \qquad \forall x \in \mathbb{R},
\end{equation}
and locally Lipschitz with polynomial growth, that is
\begin{equation} \label{Lipschitz}
|\varphi'(x)-\varphi'(y)|\leq |x-y||P(x)+P(y)|, \qquad \forall x,y \in \mathbb{R},
\end{equation} 
for a polynomial $P$. 
\end{itemize}
\vspace{0.3 cm}

\noindent
The particle system \eqref{eq:sde-part} has been originally studied in \cite{benachour1998nonlinear}. However, as pointed out in \cite{malrieu2003convergence}, the asymptotic properties of the system are rather ill-behaved in terms of uniformity and long term behaviour, and it is more appropriate to consider the projected particle system
\begin{equation} \label{Yprocess}
Y_{t}^{i,N}:=X_{t}^{i,N}-\frac{1}{N}\sum_{j=1}^{N}X_{t}^{j,N},\qquad 1\le i \le N, \qquad t \ge 0.
\end{equation}
The \textit{mean field equation}, which determines the asymptotic behaviour of the process $Y^N$ at 
\eqref{Yprocess}, is given by the $1$-dimensional McKean-Vlasov equation
\begin{equation} \label{meanfield}
\overline{X}_{t}=\overline{X}_{0}+B_{t}-\frac{1}{2}\int_{0}^{t} (\varphi'\star\mu_s )(\overline{X}_{s})\,ds,\qquad t\geq0,
\end{equation}
where $\mu_t (dx)=\P(\overline{X}_t \in dx)$ and 
\[
( \varphi'\star\mu_t )(x)=\int_{\mathbb{R}} \varphi'(x-y)\mu_t (dy), \qquad x \in \mathbb{R}, \qquad t \ge0.
\]
Under Assumption (A) the measure $\mu_t$ possesses a smooth Lebesgue density, which solves the partial differential equation
\begin{equation}
\frac{\partial}{\partial t} \mu_t = \frac 12 \frac{\partial^2}{\partial x^2} \mu_t + \frac 12 
\frac{\partial}{\partial x} ( (\varphi' \star \mu_t) \mu_t ), \qquad \mu_0 (dx)=\P(\overline{X}_{0}\in dx).
\end{equation} 
The stochastic differential equation \eqref{meanfield} admits an invariant density $\pi$, which is described by an integral equation of convolution type:
\begin{equation} \label{pidef}
\pi(x) = Z^{-1}_{\pi} \exp\left(-(\varphi \star \pi)(x) \right) \qquad \text{with} \quad 
Z_{\pi}=\int_{\mathbb{R}} \exp\left(-(\varphi \star \pi)(x) \right) dx.   
\end{equation}
In this article we will consider semiparametric estimation of the interaction function $\varphi'$ and the identity \eqref{pidef} will be key for our approach. 

Next, we will demonstrate a propagation of chaos result for the particle system \eqref{Yprocess}. We recall that the \textit{Wasserstein $p$-distance} between two probability measures $\mu_1, \mu_2$ on $\mathbb{R}$ is defined by
\[
W_p(\mu_1, \mu_2):= \Big( \inf_{X_1\sim \mu_1, X_2\sim \mu_2} \mathbb{E}[|X_1-X_2|^p] \Big)^{1/p},
\]
where the infimum is taken over all couplings $(X_1,X_2)$ such that $X_i$ has the law $\mu_i$, $i=1,2$. The following theorem has been shown in \cite{cattiaux2008probabilistic,malrieu2003convergence}.

\begin{theo}[Theorems 5.1 and 6.2 in \cite{malrieu2003convergence}] \label{th1}
Let $\overline{X}^i$, $1\le i \le N$, be i.i.d.\ copies of the process $\overline{X}$ defined at 
\eqref{meanfield} so that every $\overline{X}^{i}$ is driven by the same Brownian motion as the $i$th particle of the system \eqref{eq:sde-part} and they are equal at time $0$. Denote by 
$$\Pi_{N,T}= N^{-1} \sum_{i=1}^N \delta_{Y_{T}^{i,N}}$$ 
the empirical distribution of the projected particle system $Y_{T}^{i,N}$, $1 \le i \le N$, and by $\Pi$ the law associated to the invariant density $\pi$.  Under Assumption (A) there exist constants $C_1,C_2>0$ (independent of $N,T$) such that 
\begin{equation}
\sup_{t\geq 0} \mathbb{E}[|Y_{t}^{i,N} - \overline{X}^i_t|^2] \leq C_1 N^{-1}
\end{equation}
and
\begin{equation} \label{wasser}
\mathbb{E}[W^2_1(\Pi_{N,T}, \Pi)] \leq C_1 N^{-1} + C_2 \exp(-\lambda T)=: N_T^{-1}
\end{equation}
where the constant $\lambda>0$ has been introduced in \eqref{convex}.
\end{theo}

\noindent
The estimate \eqref{wasser} states that the invariant distribution $\Pi$ of the mean field equation 
\eqref{meanfield} is well approximated by the empirical measure $\Pi_{N,T}$ and gives the error bound associated with this approximation. In the next section we will use this result in our estimation procedure.



\section{Statistical problem and the estimation method} \label{sec3}
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We assume that the data
\[
Y_{T}^{1,N}, \ldots, Y_{T}^{N,N}
\]
is observed and $N,T \to \infty$ (and, as a consequence, $N_T \to \infty$), and our goal is to estimate 
the interaction function $\varphi'$ introduced in \eqref{eq:sde-part} in a semiparametric setting.  We remark that the sampling scheme is rather unusual as we observe the particle system only at the terminal time $T$. According to the identity  \eqref{pidef} and the statement \eqref{wasser}, the considered data suffices to identify the interaction function $\varphi'$ via the mean field limit. 
\par
Due to the complexity of the integral equation  \eqref{pidef}, which we will heavily rely on in our estimation procedure, we can not treat fully general interaction functions $\varphi'$. Instead we consider a semiparametric model of the form
\begin{equation} \label{sempar}
\varphi (y) = \sum_{0 < j \le J} a_j \varphi_j (y) + \beta (y), \qquad y \in \mathbb{R},
\end{equation}
where
\begin{equation}
\varphi_j (y) = y^{2j}, \ 0 < j \le J_1, \qquad \varphi_j (y) = \cos (\theta_j y),  \ J_1 < j  \le J, \qquad y \in \mathbb{R},
\end{equation}
for some known distinct frequencies 
$\theta_{J_1+1} >0$, \dots, $\theta_{J} >0$
and known positive integers $J_1 \le J$. 
The parameters $a_1>0$, $a_2 \ge 0$, \dots, $a_{J_1-1} \ge 0$, $a_{J_1}>0$, $a_{J_1+1} \in \mathbb{R}$, \dots, $a_J \in \mathbb{R}$
and the function $\beta$ are unknown.
The nonparametric component $\beta\in C^{2}(\mathbb{R})$ is even, bounded and such that $\beta'$ is bounded, $\| \beta' \|_{L^1(\mathbb{R})} := \int_{\mathbb{R}} |\beta'(y)| dy <\infty$ (later we will also
assume that $\| \beta'' \|_{L^2(\mathbb{R})} := (\int_{\mathbb{R}} |\beta''(y)|^2 dy)^{1/2} < \infty$).
The strict convexity condition 
\eqref{convex} is induced by the assumption
\begin{equation}
2a_1 - \sum_{J_1 < j \le J}  \theta_j^2 |a_j| + \beta''(y) \geq \lambda>0, \qquad \forall y \in \mathbb{R}.
\end{equation}
The presence of the polynomial term in \eqref{sempar}  gives upper and lower bounds for $\pi$, which are required in the proofs (and are difficult to obtain in the general setting). The presence of the trigonometric terms is for modelling purpose only and does not influence the estimation theory.

Our approach will be based upon the integral equation  \eqref{pidef}. We will first provide the representation of the invariant density $\pi$ in the setting \eqref{sempar}. In the following,
for any function $f\in L^1(\mathbb{R})$, we denote by $\mathcal{F}(f)$ the Fourier transform of $f$. 

\begin{lem} \label{lem1} 
For $\varphi$ given in \eqref{sempar}, we have
\begin{equation}
(\varphi \star \pi) (y) = \alpha_0 + \sum_{0 < j \le J} \alpha_j \varphi_j (y) + (\beta \star \pi)(y), \qquad y \in \mathbb{R},
\end{equation}
with $\alpha_0 = \sum_{0< k \le J_1} m_{2k} a_k$ and
\begin{equation}\label{def:al}
\alpha_j = \sum_{j \le k \le J_1} \binom{2k}{2j} m_{2(k-j)} a_k, \ 0 <  j \le J_1, \qquad \alpha_j = a_j \mathcal{F}(\pi) (\theta_j), \ J_1 < j \le J,
\end{equation}
where $m_k=\int_{\mathbb{R}} y^k \pi(y) dy$ denotes the $k$th moment of the invariant measure $\Pi$.
\end{lem}

\begin{proof}
We consider the sum of $a_k (\varphi_k \star \pi) (x)$ over $0<k\le J_1$, where
\begin{equation} 
(\varphi_k \star \pi)(y) = \int (y-x)^{2k} \pi(x) dx =  \sum_{j=0}^k
\binom{2k}{2j} m_{2(k-j)} y^{2j}  
\end{equation}
since $\pi$ is symmetric. Then interchanging the order of summation, we get the formula for the coefficients $\alpha_j$, $0 \le j \le J_1$. Similarly,  for $J_1<j \le J$, the coefficients $\alpha_j$ are obtained through the identity
\begin{equation}
\int \cos(\theta_j (y-x)) \pi(x) dx = \cos(\theta_j y) \int \cos(\theta_j x) \pi(x) dx,
\end{equation}
where we again have used the symmetry of $\pi$. This completes the proof of Lemma \ref{lem1}. 
\end{proof}

\noindent
We now proceed with the introduction of the estimation procedure, which consists of four steps:

\begin{itemize}
\item[(i)] Estimate the derivative of the log-density
\begin{equation} \label{lfunction}
l(y):= (\log \pi)'(y)= \frac{\pi'(y)}{\pi(y)}, \qquad y \in \mathbb{R},
\end{equation}
via a kernel-type estimator $l_{N,T}$ based on the observed data $Y_{T}^{1,N}, \ldots, Y_{T}^{N,N}$.
\item[(ii)] Estimate the parameter  $\boldsymbol{\alpha}:=(\alpha_1,\dots,\alpha_J)^\top \in \mathbb{R}^J$ using the minimum contrast method based on 
\begin{equation} 
l(y,\boldsymbol{\alpha}) = - \sum_{j=1}^J \alpha_j \varphi'_j(y),
\end{equation} 
which approximates $l(y) = l(y,\boldsymbol{\alpha}) - (\beta'\star \pi)(y)$ for large values of $y \in \mathbb{R}$.
\item[(iii)]  Use the results of step (ii) to construct an estimator $\Psi_{N,T}$ of 
\begin{equation}
\Psi(y):= -(\beta' \star \pi)(y), \qquad y \in \mathbb{R}.
\end{equation}
\item[(iv)] Finally, apply the deconvolution 
\begin{equation}
\mathcal{F} (\beta') (z)= - \frac{\mathcal{F} (\Psi )(z)}{\mathcal{F}(\pi)(z)} = - \frac{\mathcal{F} (\Psi )(z) \overline{\mathcal{F}(\pi)(z)}}{|\mathcal{F}(\pi)(z)|^2} , \qquad z \in \mathbb{R},
\end{equation}
and Fourier inversion to obtain an estimator $\beta'_{N,T}$ of $\beta'$.
\end{itemize}

\noindent
\textbf{Estimation of the function $l$.}
Following the latter we first introduce kernel estimators of $\pi$ and $\pi'$. For any function $f: \mathbb{R}\to \mathbb{R}$ and $u>0$ we use the standard notation
$f_u(x):= u^{-1} f(u^{-1} x)$. 
Let $K$ be a smooth kernel of order $m \ge 2$, that is  
\[
\int_{\mathbb{R}} K(x)dx=1, \qquad \int_{\mathbb{R}} x^{j}K(x)dx=0 \quad j=1,\ldots, m-1, \qquad  \int_{\mathbb{R}} x^{m}K(x)dx \not =0.
\]
Let $h_i = h^i_{N,T}$, $i=0,1$, be two bandwidth parameters vanishing as $N,T \to \infty$. We define 
\begin{equation} \label{kernelest}
\pi_{N,T} (y) := \frac{1}{N} \sum_{i=1}^{N} K_{h_0} \Big(y - Y_T^{i,N}\Big),
\quad
\pi'_{N,T} (y) := \frac{1}{N h_1} \sum_{i=1}^{N} K'_{h_1} \Big(y - Y_T^{i,N}\Big), \quad y \in \mathbb{R}.
\end{equation}  
Next, we introduce a threshold $\delta = \delta_{N,T} \to 0$ as $N,T\to\infty$ and set
\begin{equation} \label{lnt}
l_{N,T}(y):= \frac{\pi'_{N,T} (y)}{\pi_{N,T} (y)} \mathbbm{1}_{\{\pi_{N,T} (y)>\delta \}}, \qquad y \in \mathbb{R},
\end{equation}
which is an estimator of the function $l$ given at \eqref{lfunction}. \\ \\
\textbf{Estimation of the parametric part.} Recall the identity  $l(y) = l(y,\boldsymbol{\alpha})-(\beta'\star \pi)(y)$. Using $(\beta'\star \pi)(y) \to0$ as $|y|\to \infty$ since $\beta' \in L^1(\mathbb{R})$, we will construct the minimum contrast estimator for $\boldsymbol{\alpha}$. More specifically, we introduce an integrable weight function $w$ with support $[\epsilon,1]$ ($\epsilon\in (0,1)$) and a parameter $U = U_{N,T}\to \infty$ as $N,T\to\infty$. For $\boldsymbol{\alpha} \in \mathbb{R}^J$, we define
\begin{equation} \label{alrho}
\boldsymbol{\alpha}_{N,T} := \arg \min_{\boldsymbol{\alpha}\in \mathbb{R}^J} \int (l_{N,T} (y) - l(y,\boldsymbol{\alpha}) )^2  w_U (y) dy.
\end{equation}  
	We can use the relations \eqref{def:al} to estimate the coefficients $\boldsymbol{a} = (a_1,\dots,a_J)^\top$, based on the empirical moments $m_{2j; N,T}$, $1 < j < J_1$, 
	and the empirical Fourier moments $\mathcal F (\Pi_{N,T})(\theta_j)$, $J_1 < j \le J$, of the particle system:
	\begin{equation}
	m_{k; N,T} := \frac{1}{N} \sum_{i=1}^N (Y^{i,N}_T)^k, \ k \in \mathbb{N}, \qquad 
	\mathcal{F}(\Pi_{N,T})(z):=\frac 1N \sum_{i=1}^N \exp (\mathrm{i} z Y_T^{i,N} ), \ z \in \mathbb{R}.
	\end{equation}
	By solving the corresponding linear systems we so construct estimates $\boldsymbol{a}_{N,T}$ for the coefficients $\boldsymbol{a}$.
\\ \\
\textbf{Estimation of the nonparametric part.} Given the estimator $\boldsymbol{\alpha}_{N,T}$ introduced
in the previous step, we define
\begin{equation}
\Psi_{N,T}(y) = ( l_{N,T} (y) - l (y, \boldsymbol{\alpha}_{N,T}) ) \mathbbm{1}_{\{|y|\leq \epsilon U\}}, \qquad y \in \mathbb{R},
\end{equation}
which provides an estimator of the function $\Psi = -\beta' \star \pi$. In the next step we choose another threshold $\omega = \omega_{N,T} \to 0$ and introduce 
\begin{equation}
\mathcal{F}(\beta'_{N,T}) (z):= -\frac{\mathcal{F}(\Psi_{N,T}) (z) \overline{\mathcal{F}(\Pi_{N,T})(z)} }{|\mathcal{F}(\Pi_{N,T}) (z)|^2} \mathbbm{1}_{\{|\mathcal{F}(\Pi_{N,T})(z)|>\omega \}}, \qquad z \in \mathbb{R}.
\end{equation}
Finally, we use the Fourier inversion to estimate the function $\beta'$:
\begin{equation}\label{def:h}
\beta'_{N,T}(y):= \frac{1}{2\pi} \int \exp(-\mathrm{i} zy) \mathcal{F}(\beta'_{N,T})(z) dz, \qquad y \in \mathbb{R}.
\end{equation}
In the following we will derive asymptotic properties of all estimators introduced in this section.



\section{The asymptotic theory} \label{sec4}
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We start our asymptotic analysis with the estimator $l_{N,T}$. In the following the bandwidth and threshold parameters are chosen as
\begin{equation} \label{bandwidth}
h_0 = N_T^{-\frac{1}{2(m+1)}}, \qquad h_1 = N_T^{-\frac{1}{2(m+2)}}, \qquad \delta = \delta_0 \exp (- \bar \alpha_1 U^{2J_1} ),
\end{equation} 
where $N_T$ is the rate introduced in \eqref{wasser}, $m$ is the order of the kernel $K$ and
$\delta_0 := (2 Z_\pi )^{-1} \exp ( - \alpha_0 - \sum_{J_1 < j \le J} |\alpha_j| - \| \beta \|_\infty)$, 
$\bar \alpha_1 :=\sum_{0 < j \le J_1} \alpha_j$.
Here and in what follows, $\| f \|_\infty := \sup_{y \in \mathbb{R}} |f(y)|$ for $f : \mathbb{R} \to \mathbb{R}$. Furthermore, we write $a_n \lesssim b_n$ when there exists a constant $C>0$, independent of $n$, such that $a_n\leq Cb_n$. Our first result is the following proposition.






\begin{prop} \label{prop1}
Let $\delta$, $h_i$, $i=0,1$, be defined as in \eqref{bandwidth} and $U \ge 1$. Then
\begin{equation}
\sup_{|y|\leq U}  \mathbb{E} \left[ |l_{N,T}(y) - l(y)|^2 \right]^{\frac 1 2} \lesssim \exp ( \bar \alpha_1 U^{2J_1} ) \left(
N_T^{-\frac{m}{2(m+2)}} + U^{2J_1-1} N_T^{-\frac{m}{2(m+1)}} \right).
\end{equation}
\end{prop}
\begin{proof} See Section  \ref{sec6}.
\end{proof}

\noindent
We observe that the upper bound in Proposition \ref{prop1} grows exponentially in $U$, which will strongly affect convergence rates for all parameters of the model.
We will now find the explicit expression for the estimator of $\boldsymbol{\alpha}$. 
For this purpose, in \eqref{lnt}  we write $l (y,\boldsymbol{\alpha}) = \boldsymbol{l}(y/U)^\top \cdot \boldsymbol{\alpha}^{U}$, where
\begin{equation}
\boldsymbol{l} (y) = -(\varphi'_1 (y), \dots,\varphi'_{J_1} (y), \varphi'_{J_1+1}(Uy), \dots, \varphi'_J(Uy))^\top, \qquad y \in \mathbb{R},
\end{equation}
and
\begin{equation}
\boldsymbol{\alpha}^{U} = (\alpha_1 U, \alpha_2U^3, \dots, \alpha_{J_1} U^{2J_1-1}, \alpha_{J_1+1}, \alpha_{J_1+2}, \dots, \alpha_J )^\top.
\end{equation}
Then the unknown $\boldsymbol{\alpha}^{U}$ and analogously scaled estimator $\boldsymbol{\alpha}_{N,T}^{U}$ satisfy the identities
\begin{equation}
Q \boldsymbol{\alpha}_{N,T}^{U} = \int l_{N,T} (y) \boldsymbol{l} (y/U) w_U(y) dy, \qquad Q \boldsymbol{\alpha}^{U} = \int (l(y)+(\beta' \star \pi)(y)) \boldsymbol{l} (y/U) w_U(y) dy,
\end{equation}
where 
\begin{equation} \label{Qdef}
Q:= \int \boldsymbol{l} (y) \boldsymbol{l} (y)^\top w (y) d y \in \mathbb{R}^{J\times J}.
\end{equation}
Notice that the components of the function $\boldsymbol{l}$ are linearly independent on any 
interval $[s,t]$, $s<t$, since
$U\theta_{J_1+1},\dots, U\theta_{J}$ are all distinct. 
In this case the matrix $Q$ is positive definite and hence invertible. 

In the next proposition we derive convergence rates for the estimates $\boldsymbol{\alpha}_{N,T}^{U}$
and $\Psi_{N,T}$. By $\| \cdot \|_2$ we denote the Euclidean norm.


\begin{prop}\label{prop2}
	Let $\delta$, $h_i$, $i=0,1$, be defined as in \eqref{bandwidth} and $U \ge 1$.  Then
	\begin{align*}
	&\mathbb{E} \left[ \| \boldsymbol{\alpha}_{N,T}^{U}- \boldsymbol{\alpha}^{U} \|^2_2\right]^{\frac 1 2} \lesssim \exp (\bar \alpha_1 U^{2J_1}) 
	\left(N_T^{-\frac{m}{2(m+2)}} + U^{2J_1-1} N_T^{-\frac{m}{2(m+1)}} \right) \\ 
	&\qquad\qquad+ \frac{\exp (- \alpha_{J_1} (\epsilon U/2)^{2J_1})}{(\epsilon U/2)^{2J_1}} + U^{-1} \int_{|y|>\epsilon U/2} |\beta' (y)| d y,\\
	& \mathbb{E} \left[ \int_{\mathbb{R}} | \Psi_{N,T} (y) - \Psi (y) |^2 d y \right]^{\frac 1 2}\lesssim \exp (\bar \alpha_1 U^{2J_1}) U^{\frac 1 2}    \left(N_T^{- \frac{m}{2(m+2)}} + U^{2J_1-1} N_T^{-\frac{m}{2(m+1)}} \right)\\ 
	&\qquad\qquad + \frac{\exp(-\alpha_{J_1} (\epsilon U/2)^{2J_1})}{(\epsilon U/2)^{2J_1-1}} + U^{-\frac{1}{2}} \int_{|y|>\epsilon U/2} |\beta' (y)| d y+\left( \int_{|y|>\epsilon U/2} |\beta'(y)|^2 dy \right)^{\frac{1}{2}}.
	\end{align*}
\end{prop}
\begin{proof} See Section  \ref{sec6}.
\end{proof}

\noindent
We remark that the convergence rates for the parametric part of the model are logarithmic, which is rather unusual for parametric estimation problems.  
Indeed, the logarithmic rate is obtained under a proper choice of $U$ when there exists $q>0$ such that $0<\liminf_{y \to \infty} F_{q,\beta'} (y) \le \limsup_{y \to \infty} F_{q,\beta'} (y) <\infty$, where $F_{q,\beta'}(y)=y^q \int_y^\infty |\beta'(u)| du$, $y>0$.
We believe that the reason for such a slow convergence rate is a convolution type structure of the invariant density $\pi$.


\begin{rem} \label{rem1} \rm
The statement of Proposition \ref{prop2} can be transferred from $\boldsymbol{\alpha}$ to the original parameter $\boldsymbol{a}$ via the identities \eqref{def:al}. 
This follows from the estimate $\mathbb{E} [ |m_{2j; N,T} - m_{2j}|^{2k}]  \lesssim N_T^{-1}$ when $X_0^1$ has sufficiently high moments, see e.g.\ \cite{cattiaux2008probabilistic,malrieu2003convergence}. \qed
\end{rem}

\noindent
It is evident from Proposition \ref{prop2}  that the rate of convergence for $\Psi_{N,T} $ crucially depends on the tail behaviour of the function $\beta'$.  
The next corollary, which is an immediate consequence of the previous result, gives the precise bound. 
 
\begin{cor}\label{cor1}
	 In the setting of Proposition \ref{prop2}  assume that there exists $p>0$ such that $\liminf_{y \to \infty} \Phi_{p,\beta'} (y)>0$, where 
	 \begin{equation}\label{def:Phip}
	 \Phi_{p,\beta'} (y):= y^{p-(1/2)} \int_y^\infty |\beta'(u)| du + y^p \Big( \int_y^\infty |\beta'(u)|^2 du \Big)^{1/2}, \qquad y>0.
	 \end{equation}
	Choose $U = (c \log N_T)^{1/(2J_1)}$ for some $0< c < m / (2 (m+2) \bar \alpha_1)$. Then 
	\begin{equation}
	\mathbb{E} \left[ \int_{\mathbb{R}} | \Psi_{N,T} (y) - \Psi (y) |^2 d y \right] \lesssim (\log N_T)^{-p/J_1} \Phi^2_{p,\beta'} ( (c \log N_T )^{1/(2J_1)} \epsilon/2 ).
	\end{equation}
	Moreover, if $\limsup_{y \to \infty}\Phi_{p,\beta'} (y)<\infty$, then
	\begin{equation} \label{psiconv}
	\mathbb{E} \left[ \int_{\mathbb{R}} |\Psi_{N,T} (y)-\Psi (y)|^2 d y \right]\lesssim (\log N_T)^{-p/J_1}.
	\end{equation}
\end{cor}

\noindent
We see that the convergence rate in  \eqref{psiconv} depends on the nonparametric part of the model through the parameter $p$ and on the highest degree polynomial in the parametric part through $J_1$ (in contrast, the trigonometric part of the model does not influence the convergence rate). 
This phenomenon will translate to the estimation of $\beta'$.










The following proposition will play a crucial role for the analysis of the estimator $\beta'_{N,T}$.

\begin{prop}\label{prop3}
	In the setting of Proposition \ref{prop2} assume $\beta'' \in L^2(\mathbb{R})$. Then
	\begin{align}
	\mathbb{E}\left[\int_{\mathbb{R}} | \beta'_{N,T}(y)-\beta' (y) |^2 d y \right]  &\lesssim \omega^{-2}  \Big(  \mathbb{E}\left[\int_{\mathbb{R}} | \Psi_{N,T}(y)-\Psi(y) |^2 d y \right] + N_T^{-1} \Big)\nonumber\\ 
	&\qquad + N_T^{-1} \int_{|\mathcal{F}(\pi) (z)| > 2\omega} |\mathcal{F}(\beta') (z)|^2 |\mathcal{F}(\pi)(z)|^{-2} z^2 d z\\ 
	&\qquad+ \int_{|\mathcal{F}(\pi) (z)| \le 2 \omega} |\mathcal{F}(\beta') (z)|^2 d z.\label{int:betaMISE}
	\end{align}
\end{prop}
\begin{proof} See Section  \ref{sec6}.
\end{proof}

\noindent
We observe that, up to the presence of the factor $\omega^{-2}\to \infty$ which can be chosen arbitrarily, the convergence rate for the nonparametric part 
$\beta'$ is transferred from Proposition \ref{prop2}. Indeed, the next result shows that the second and the third terms in \eqref{int:betaMISE} are negligible
under appropriate assumption on $\beta'$. 


\begin{prop}
	\label{prop:rate-entire}
	In the setting of Proposition \ref{prop3} assume that $\beta'$ is an entire function of the first order and type less than $\vartheta>0$, i.e.
	\begin{equation}
	|\beta'(z)| \le A \exp (  \vartheta |z| ), \qquad z \in \mathbb{C}, 
	\end{equation}
	with $A >0$ and $\vartheta \le \lambda^{1/2}$. 
	Let $\liminf_{y \to \infty} \Phi_{p,\beta'} (y)>0$ for $p>0$.
	Choose $U = (C \log N_T)^{1/(2J_1)}$ for some $0< C < m/(2(m+2) \bar \alpha_1)$. Then 
	\begin{equation}\label{MISEbeta}
	\mathbb{E}\left[ \int_{\mathbb{R}} |\beta'_{N,T} (y) -\beta' (y)|^2 d y\right] \lesssim \omega^{-2} (\log N_T)^{-p/J_1} \Phi_{p,\beta'}^2 ( ( C \log N_T )^{1/(2J_1)} \epsilon/2 ).
	\end{equation}
	Moreover, if $\limsup_{y \to \infty} \Phi_{p,\beta'} (y)>0$, then
	\begin{equation}
	\mathbb{E}\left[\int_{\mathbb{R}} | \beta'_{N,T}(y)-\beta' (y) |^2 d y \right]  \lesssim \omega^{-2} (\log N_T)^{-p/J_1}.
	\end{equation}
\end{prop}
\begin{proof} See Section  \ref{sec6}.
\end{proof}

\begin{ex}
	As an example of functions satisfying the conditions of Proposition \ref{prop:rate-entire} we may consider 
	\begin{equation}
	\beta_1(y) = \frac{1-\cos(b y)}{y^2}, \ b>0, \qquad \beta_2(y) = \frac{\sin^{2k}(y)}{y^{2k}}, \ k \in \mathbb{N}.
	\end{equation}
	One can easily check that both function fulfil the necessary assumptions with $p=3/2$
	in case of $\beta_1$ and 
	$p=2k-1/2$
	in case of $\beta_2$.
	\qed
\end{ex}

\section{Lower bound for the nonparametric component} \label{sec5}
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\renewcommand{\theequation}{\thesection.\arabic{equation}}

In this section we will derive the lower bound for the estimation of the nonparametric component $\beta'$. We consider the simple model with i.i.d.\ observations $Y_1, \dots, Y_N$ having the following density:
\begin{equation} \label{iidmodel}
\pi_{\beta}(y)= Z_{\pi_{\beta}}^{-1} \exp \Big(- \Big( \Big( \sum_{j=1}^J a_j \varphi_j + \beta \Big) \star \pi_{\beta} \Big) (y) \Big), \qquad y \in \mathbb{R},
\end{equation}
where
\begin{equation} 
Z_{\pi_\beta} =  \int_{\mathbb{R}} \exp \Big(- \Big( \Big( \sum_{j=1}^J a_j \varphi_j + \beta \Big) \star \pi_{\beta} \Big)(y) \Big) dy
\end{equation}
and $\varphi_j (y) = y^{2j}$, $y \in \mathbb{R}$, $1 \le j \le J$, for given $J \ge 1$. We assume that the constants $a_1 >0$, $a_2 \ge 0$, \dots, $a_{J-1} \ge 0$, $a_J>0$ are known, the function $\beta(y)$ is even and such that $\varphi''(y) \ge \lambda$, $y \in \mathbb{R}$, with known $\lambda >0$.
We remark that the nonparametric estimation problem is similar in spirit to the classical deconvolution problem, but we can not apply the same techniques to derive the lower bound. 
\par
We consider the following class of functions 
\begin{align*}
\mathcal F_{p,C, C_0, \boldsymbol{a},\lambda} := \Big\{\Big.f \in C_b^{2}(\mathbb{R}): ~&\|f\|_{\infty} \le C_0,~\|f' \|_\infty \le C_1,~\inf_{y \in {\mathbb{R}}} f''(y)\geq  -C_2\\
\qquad\qquad\qquad &\limsup_{y \to \infty} y^{2p} \int_y^\infty |f'(u)|^2 du \le C \Big\}\Big.,
\end{align*}
where $C_b^2(\mathbb{R})$ denotes the space of twice continuously differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that
$f,f',f''$ are bounded. Moreover, $C,C_0>0$ and
\begin{equation}
C_1 := \lambda^{1/2} \Big(1 - \sum_{1 < j \le J} 2j c_j a_j / \lambda^j \Big)>0, \qquad C_2 := 2a_1-\lambda>0,
\end{equation} 
where
$c^2_j := 2 (2(2j-1))! / (2j-1)!$, $1<j\le J$. We remark that the condition $C_1>0$ holds whenever $\lambda>0$ is sufficiently large. 
We also note that assumption (A) is automatically satisfied when $\beta \in \mathcal F_{p,C, C_0, \boldsymbol{a},\lambda}$. The main result of this section is the minimax bound over the functional class $\mathcal F_{p,C, C_0, \boldsymbol{a},\lambda}$.

\begin{theo} \label{minimax}
	Denote the law of $Y_1$ by $\Pi_{\beta}$, and consider the functional class $\mathcal F_{p,C, C_0, \boldsymbol{a},\lambda}$ for 
	$p>1/2$. Then there exists a constant $c_0>0$ (depending on $p$, $C$, $C_0$, $\boldsymbol{a}$, $\lambda$) such that
	\begin{equation} \label{minimaxrate}
	\inf_{\beta'_N} \sup_{\beta \in \mathcal F_{p,C, C_0, \boldsymbol{a},\lambda}} \Pi_{\beta}^{\otimes N}\Big( \|\beta'_N - \beta'\|_{L^2(\mathbb{R})}^2 >c_0 (\log N)^{-p/J} \Big)>0.
	\end{equation}
\end{theo}
\begin{proof} See Section  \ref{sec6}.
\end{proof}

\noindent
We remark that our estimator $\beta'_{N,T}$ matches the lower bound up to the factor $\omega^{-2}$, which can be chosen to diverge to $\infty$ at an
arbitrary slow rate. 

\begin{rem} \label{rem2} \rm
The result of Theorem \ref{minimax} can be compared to a classical deconvolution problem. Consider a model
\[
Y_i=X_i +\varepsilon_i, \qquad i=1,\ldots, N,
\]
where $(X_i)_{i\ge 1}$ and $(\varepsilon_i)_{i\ge 1}$ are mutually independent i.i.d.\ sequences. Assume that $X_1$ (resp. $\varepsilon_1$) has a Lebesgue density $f$
(resp. $g$), and we are in the super smooth setting, that is,
\[
\mathcal F(g)(z) \sim \exp(-\text{const} \cdot |z|^{2J}), \quad |z|\to \infty.
\] 
When the density $f$ satisfies the condition $\int_{\mathbb{R}} |\mathcal F(f)(z)|^2 |z|^{2p} dz \leq C$, $p>1/2$, it is well known that the minimax rate for the estimation of the density $f$ becomes $(\log N)^{-p/J}$ (see e.g. Theorem 2.14(b) in \cite{M09}). While in the classical deconvolution problem the assumptions are imposed in the Fourier domain, we have comparable assumptions on the functions themselves. 
Notice that due to the structure of the model $\pi$ plays the role of the noise density and
the condition on  $\mathcal F(g)$ can be compared to the decay of $\pi_{\beta}$, which is determined by the leading polynomial of degree $J$. On the other hand, the integral condition on $\mathcal F(f)$ is related to the corresponding tail condition on $\beta'$. 
\end{rem}



\section{Conclusions and outlook} \label{secOut}
In this work we study the problem of estimating the drift of a MV-SDE based on observations of the corresponding particle system. We propose a kernel-type estimator and provide theoretical analysis of its convergence. 
In particular, for the nonparametric part of the model, we derive minimax convergence rates and show rate-optimality of our proposed estimator.
As a promising future research direction, one can consider the case of continuous time observations and high-dimensional MV-SDEs with general form of the drift function. In another direction the problem of estimating diffusion coefficient remains completely open.







\section{Proofs} \label{sec6}
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\renewcommand{\theequation}{\thesection.\arabic{equation}}

\subsection{Preliminary results}

\begin{lem}\label{lem2}
	Set $\bar \varphi_1 (y): = \sum_{0 < j \le J_1} \alpha_j \varphi_j(y)$, $y\in\mathbb{R}$. Then $\pi \in C^\infty (\mathbb{R})$ and for every $n$, 
	\begin{equation}
	|\pi^{(n)} (y)| \lesssim (1+|\bar \varphi'_1 (y)|)^n \exp (- \bar \varphi'_1 (y)), \qquad y\in\mathbb{R}.
	\end{equation}

\end{lem}

\begin{proof}
	We decompose $\bar \varphi = \varphi \star \pi$ as $\bar \varphi = \alpha_0 + \bar \varphi_1 + \bar \varphi_2 + \beta \star \pi$. Here $\bar \varphi_2: = \sum_{J_1 < j \le J} \alpha_j \varphi_j$ is bounded, and $\| \beta \star \pi \|_\infty \le \| \beta \|_\infty \| \pi \|_{L^1(\mathbb{R})} < \infty$. Hence, we obtain
	\begin{equation}
	\pi (y) = Z_\pi^{-1} \exp (- \bar \varphi (y)) \lesssim \exp (- \bar \varphi_1 (y)), \qquad y \in \mathbb{R}.
	\end{equation}
	We now consider its derivative. Since $\beta$ has a bounded derivative,  so does $\beta \star \pi$. We obtain
	$\pi' = - \bar \varphi' \pi $, where 
	$\bar \varphi' = \bar \varphi'_1 + \bar \varphi'_2 + \beta'\star \pi$ satisfies
	\begin{equation}
	| \bar \varphi' (y) | \lesssim 1+|\bar \varphi'_1(y)|, \qquad y \in \mathbb{R}.
	\end{equation}
	That is, statement of the proposition holds for $n=1$. 	





	For $n \ge 1$ it follows by induction using
	\begin{equation}
	\pi^{(n+1)} = (\pi')^{(n)}= - \sum_{k=0}^{n} \binom{n}{k} \bar \varphi^{(k+1)} \pi^{(n-k)} 
	\end{equation}
	with $\bar \varphi^{(k+1)} = \bar \varphi_1^{(k+1)}+\bar \varphi_2^{(k+1)}+(\beta'\star \pi)^{(k)}$, where $(\beta'\star \pi)^{(k)} = \beta' \star \pi^{(k)}$ is bounded when $\| \pi^{(k)} \|_\infty < \infty$, $\| \beta' \|_{L^1(\mathbb{R})} < \infty$.

\end{proof}

\begin{lem}\label{mom}
	Moments of the density $\pi$ in \eqref{pidef} satisfy 
	\begin{equation}
	m_{2k} \le (2k-1)!! / \lambda^k, \qquad k \in \mathbb{N}.
	\end{equation}
\end{lem}

\begin{proof}
	Set
	\begin{equation}
	I_k := \int_{\mathbb{R}} y^{2k-1} (\varphi' \star \pi) (y) \pi (y) dy. 
	\end{equation}
	Since $\varphi(y)$ is even, we have $\varphi'(y)= -\varphi'(-y)$, implying
	\begin{equation}
	2I_k = \int \int (y^{2k-1}-x^{2k-1}) \varphi' (y-x) \pi (x) \pi (y) dx dy.
	\end{equation}
	By the mean value theorem we conclude that
	\begin{equation} 
	\varphi'(y-x) = \varphi'(y-x) - \varphi'(0) = (y-x) \varphi''(z) 
	\end{equation} 
	for some $z \in (x,y)$, and by the convexity assumption,
	\begin{equation}
	2I_k \ge \lambda \int \int (y^{2k-1}-x^{2k-1})(y-x) \pi (x) \pi (y) dx dy = 2 \lambda m_{2k}.
	\end{equation}
	On the other hand,
	\begin{equation} 
	I_k = - \int y^{2k-1} d \pi (y) = (2k-1) \int y^{2(k-1)} \pi (y) d y = (2k-1) m_{2(k-1)}.
	\end{equation}
	We conclude that $m_{2k} \le m_{2(k-1)} (2k-1)/\lambda$. Induction provides the desired result. 
\end{proof}



\subsection{Proof of Proposition \ref{prop1}}




	Using
	\begin{equation}
	|l_{N,T} (y)-l(y)| \le \Big| \frac{\pi'_{N,T}(y)}{\pi_{N,T}(y)} - \frac{\pi'(y)}{\pi_{N,T}(y)} \Big| + \Big| \frac{\pi'(y)}{\pi_{N,T}(y)} - \frac{\pi'(y)}{\pi(y)} \Big|,
	\end{equation}
	on $\pi_{N,T} (y) > \delta$, we get
	\begin{align*}
	|l_{N,T} (y)-l(y)| &\le  \delta^{-1} |\pi'_{N,T}(y)-\pi'(y)| + \delta^{-1} |l(y)| |\pi(y)-\pi_{N,T}(y)| + |l(y)| \mathbbm{1}_{\{ \pi_{N,T} (y) \le \delta\}}.
	\end{align*}
	Here using $|l(y)| \le 2 \sum_{0 < j \le J_1} j \alpha_j |y|^{2j-1} + \sum_{J_1 < j \le J} \theta_j | \alpha_j | + \|\beta'\|_\infty$
	we get
	\begin{equation}
	|l(y)| \lesssim U^{2J_1-1}
	\end{equation}
	for all $|y| \le U$. Since $|(\varphi \star \pi)(y)| \le \sum_{0 \le j \le J_1} \alpha_j y^{2j} + \sum_{J_1 < j \le J} |\alpha_j| + \| \beta \|_\infty$, the chosen $\delta$ satisfies
	\begin{equation}
	2 \delta \le Z_\pi^{-1}  \exp ( - (\varphi \star \pi) (y) )  = \pi (y)
	\end{equation}
	for all $|y| \le U$. Hence, it follows that for all $|y| \le U$, 
	\begin{align*}
	\P ( \pi_{N,T} (y) \le \delta ) 
	&= \P ( \pi (y) -\pi_{N,T} (y) \ge \pi (y)- \delta )\\ 
	&\le \P ( \| \pi -\pi_{N,T} \|_\infty \ge \delta ) \le \delta^{-2} \mathbb{E} [\| \pi - \pi_{N,T} \|_\infty^2].
	\end{align*}
	Now, with $\pi_{N,T}(y) = (K_{h_0} \star \Pi_{N,T}) (y)$, we have 
	\begin{align*}
	&| (K_{h_0} \star (\Pi_{N,T} - \Pi)) (y) | 

	\le \operatorname{Lip}(K_{h_0} (y - \cdot )) W_1 (\Pi_{N,T}, \Pi)  = h_0^{-1} \operatorname{Lip}(K)  W_1 (\Pi_{N,T}, \Pi),
	\end{align*}
	where
	$\mathbb{E} [W_1^2 (\Pi_{N,T},\Pi)] \le N_T^{-1}$ due to  \eqref{wasser}.
	Furthermore, we have
	\begin{equation}
	(K_{h_0} \star \Pi) (y) - \pi (y) = \int K (x) (\pi (y+xh_0) - \pi (y)) dx,
	\end{equation}
	where $\pi \in C^\infty (\mathbb{R})$ satisfies
	$|\pi^{(n)} (y) | \lesssim (1+|y|^{2J_1-1}) \exp (-\alpha_{J_1} y^{2J_1})$, $y\in \mathbb{R}$, $n \in \mathbb{N}$, by Lemma~\ref{lem2}.

	Using a Taylor expansion of $\pi$ and that the kernel $K$ is of order $m$, we obtain
	\begin{equation}
	(K_{h_0} \star \Pi) (y) - \pi (y) = \int K(x) R_{m-1} (y+xh_0) dx,
	\end{equation}
	where
	\begin{equation}
	|R_{m-1}(y+xh_0)| \lesssim |x|^m h_0^m
	\end{equation} 
	and so
	\begin{equation}
	|(K_{h_0} \star \Pi)(y) - \pi (y)| \lesssim h_0^m
	\end{equation}
	uniformly in $y \in \mathbb{R}$.

	Similarly, 
	\begin{equation}
	| h_1^{-1} ( K'_{h_1} \star \Pi ) (y) - \pi' (y) | \lesssim h_1^m,
	\end{equation}
	because
	\begin{equation}
	h_1^{-1} ( K'_{h_1} \star \Pi ) (y)  = ( (K_{h_1})' \star \Pi ) (y) 

	= \int K(x) \pi' (y+xh_1) dx.

	\end{equation}
	We thus deduce that
	\begin{equation}
	\mathbb{E} [\| \pi_N - \pi \|_\infty^2] \lesssim N_T^{-1}h_0^{-2} + h_0^{2m}, \qquad \mathbb{E}[ \| \pi'_N - \pi' \|_\infty^2] \lesssim N_T^{-1} h_1^{-4} + h_1^{2m},
	\end{equation}
	where our choice of $h_0, h_1$ yields the optimal rate in upper bounds:
	\begin{equation}
	\mathbb{E}[ \| \pi_N - \pi \|^2_\infty] \lesssim N_T^{-\frac{m}{m+1}}, \qquad  \mathbb{E}[ \| \pi'_N - \pi' \|^2_\infty] \lesssim N_T^{-\frac{m}{m+2}}.
	\end{equation}




\subsection{Proof of Proposition \ref{prop2}}
	Recall the definition of the matrix $Q \in \mathbb{R}^{J\times J}$ introduced at \eqref{Qdef}. Since $Q$ is invertible, we deduce that
	\begin{equation}
	\mathbb{E} \left[ \| \boldsymbol{\alpha}^{U}_{N,T} -\boldsymbol{\alpha}^{U} \|_2^2 \right]^{\frac{1}{2}} 
	\le \int  \mathbb{E} [|l_{N,T}(y)-l(y)-(\beta' \star \pi)(y)|^2 ]^{\frac{1}{2}} \tilde w_U (y) d y
	\end{equation}
	where   $\tilde w (y):= \| Q^{-1} \boldsymbol{l}(y) \|_2 |w(y)|$, $y\in\mathbb{R}$. Moreover, $\|\tilde w\|_{L^1(\mathbb{R})} < \infty$
	and 
	$\|\tilde w\|_\infty < \infty$ are uniformly bounded in $U$. Hence,
	\begin{equation}
	 \mathbb{E} \left[ \| \boldsymbol{\alpha}^{U}_{N,T} -\boldsymbol{\alpha}^{U} \|_2^2 \right]^{\frac{1}{2}}
	\lesssim \sup_{|y| \le U}  \mathbb{E}[ |l_{N,T}(y) - l(y)|^2 ]^{\frac{1}{2}} + U^{-1} \int_{\epsilon U}^\infty |(\beta' \star \pi) (y)| d y.
	\end{equation}
	As for the last term, we have
	\begin{align*}
	&\int_{\epsilon U}^\infty | (\beta' \star \pi) (y) | d y \le \int_{\epsilon U}^\infty \int |\beta' (y-x)| \pi (x) d x d y\\
	&\qquad= \Big(\int_{-\infty}^{\epsilon U/2} + \int_{\epsilon U/2}^\infty \Big)
	\Big( \int_{U \epsilon-x}^\infty |\beta' (y)| d y \Big) \pi (x) d x \le \int_{\epsilon U/2}^\infty |\beta' (y)| dy + \| \beta' \|_{L^1(\mathbb{R})} \int_{\epsilon U/2}^\infty \pi (x) d x.
	\end{align*}
	Finally, note that 
	\begin{equation}
	\int_{u}^\infty \pi (x) dx \lesssim \int_u^\infty \pi_1 (x) dx,
	\end{equation}
	where $\pi_1 (x) = \exp(- \alpha_{J_1} x^{2J_1})$ satisfies $\pi'_1 (x) = -2 J_1 \alpha_{J_1} x^{2J_1-1} \pi_1 (x)$ and so, 
	\begin{equation}
	\int_u^\infty \pi_1 (x) d x \le \frac{1}{u^{2J_1-1}} \int_u^\infty x^{2J_1-1} \pi_1 (x) d x = \frac{\pi_1 (u)}{2 J_1 \alpha_{J_1} u^{2J_1-1}} .
	\end{equation}
	
	Now we consider the bound for 
	\begin{equation}
	\Psi_{N,T} (y) = ( l_{N,T} (y) - l (y,\boldsymbol{\alpha}_{N,T}) ) \mathbbm{1}_{\{|y| \le \epsilon U\}}, \qquad y \in \mathbb{R}.
	\end{equation}
	with
	\begin{equation}
	\Psi (y) = l(y) - l (y,\boldsymbol{\alpha}) = - (\beta'\star \pi)(y), \quad y\in \mathbb{R}.
	\end{equation}
	We have 
	\begin{equation}
	\int_{|y| \le \epsilon U} \mathbb{E} [|\Psi_{N,T}(y) -\Psi(y)|^2] d y \le \epsilon U\sup_{|y| \le \epsilon U} \mathbb{E} [|\Psi_{N,T}(y)-\Psi(y)|^2],
	\end{equation}
	where
	\begin{equation}
	\sup_{|y| \le \epsilon U} \mathbb{E}[ |\Psi_{N,T} (y)-\Psi(y)|^2]
	\lesssim \sup_{|y|\le \epsilon U} \mathbb{E} [|l_{N,T}(y)-l(y)|^2] + \mathbb{E}[ \| \boldsymbol{\alpha}^{U}_{N,T} - \boldsymbol{\alpha}^{U} \|^2_2].
	\end{equation}
	Finally, we deduce
	\begin{align*}
	\Big( \int_{|y|>\epsilon U} |\Psi (y)|^2 dy \Big)^{\frac{1}{2}} 
	&\le \int \Big( \int_{|y|>U\epsilon} |\beta' (y-x)|^2 dy \Big)^{\frac{1}{2}} \pi (x) d x\\

	&\le \Big( \int_{|y|>\frac{\epsilon}{2}U} |\beta' (y)|^2 dy \Big)^{\frac{1}{2}} + \| \beta' \|_{L^2(\mathbb{R})} \int_{|x|>\frac{\epsilon}{2}U}  \pi (x) d x,
	\end{align*}
	which completes the proof of Proposition \ref{prop2}. 
	

\subsection{Proof of Proposition \ref{prop3}}
	Define a function $\beta_{N,T}^*$ via the formula
	\begin{equation}
	\mathcal{F}(\beta^\ast_{N,T}) (z) = \mathcal{F}(\beta') (z) \mathbbm{1}_{\{|\mathcal{F}(\Pi_{N,T})(z)|>\omega\}}.
	\end{equation}
	Write $\mathcal{F}(\Pi) = \mathcal{F}(\pi)$. Use the identity
	\begin{align*}
	&\mathcal{F}(\beta'_{N,T}-\beta^\ast_{N,T}) (z)\\  
	&\qquad= (- \mathcal{F}(\Psi_{N,T}-\Psi) (z) + \mathcal{F}(\beta') (z) \mathcal{F} (\Pi-\Pi_{N,T})(z) ) \frac{\overline{\mathcal{F}(\Pi_{N,T}) (z)} }{|\mathcal{F}(\Pi_{N,T}) (z)|^2} \mathbbm{1}_{\{|\mathcal{F}(\Pi_{N,T})(z)|>\omega\}},
	\end{align*}
	where by the Kantorovich--Rubinstein dual formulation, we have
	\begin{equation}
	|\mathcal{F}(\Pi-\Pi_{N,T}) (z)| \le |z| W_1 (\Pi_{N,T},\Pi).
	\end{equation}
	As a result,
	\begin{align*}
	 \mathbb{E} \left[\int_{\mathbb{R}} | \beta'_{N,T}(y)- \beta^\ast_{N,T} (y) |^2 d y \right]^{1/2} &\le \omega^{-1}   \mathbb{E} \left[  \int_{\mathbb{R}} | \Psi_{N,T} (y) - \Psi (y)|^2 d y \right]^{1/2}\\ 
	&\qquad+ \omega^{-1} \Big( \mathbb{E} [W_1^2 (\Pi_{N,T}, \Pi)] \int_{\mathbb{R}} |\beta'' (y)|^2 d y \Big)^{1/2}.
	\end{align*}
	Furthermore, it holds that
	\begin{align*}
	&\mathbb{E} \left[ \int_{\mathbb{R}} |\beta^*_{N,T} (y) - \beta' (y) |^2 d y \right]\\ 
	&\qquad= \frac{1}{2 \pi} \Big( \int_{|\mathcal{F}(\pi) (z)| > 2\omega} + \int_{|\mathcal{F}(\pi) (z)| \le 2 \omega} \Big) |\mathcal{F}(\beta') (z)|^2 \P ( |\mathcal{F}(\Pi_{N,T}) (z)| \le \omega ) d z.
	\end{align*}
	If $\omega<|\mathcal{F}(\pi) (z)|/2$, then we have by the Markov and Jensen's inequalities
	\begin{align*}
	\P ( |\mathcal{F}(\Pi_{N,T}) (z) | \le \omega ) &\le \P (|\mathcal{F}(\Pi_{N,T}-\Pi) (z)| \ge |\mathcal{F}(\pi) (z)| - \omega  )\\ 
	&\le \frac{\mathbb{E}[ |\mathcal{F}(\Pi_{N,T}-\Pi) (z)|^2]}{(|\mathcal{F}(\pi)(z)| - \omega)^2}.
	\end{align*}
	Consequently, we obtain
	\begin{align*}
	\mathbb{E} \left[ \int_{\mathbb{R}} |\beta^*_{N,T} (y) - \beta' (y) |^2 d y \right] &\le  \frac{1}{2 \pi } \Big( \int_{|\mathcal{F}(\pi) (z)| > 2\omega}  |\mathcal{F}(\beta') (z)|^2 \frac{\mathbb{E} [W_1^2(\Pi_{N,T},\Pi)]|z|^2 }{|\mathcal{F}(\pi) (z)|^2/4} d z\\ 
	&\qquad
	+ \int_{|\mathcal{F}(\pi) (z)| \le 2 \omega}  |\mathcal{F}(\beta') (z)|^2 d z\Big),
	\end{align*}
	which completes the proof of Proposition \ref{prop3}. 

\subsection{Proof of Proposition \ref{prop:rate-entire}}
		We use Proposition \ref{prop2}. It suffices to show that on the r.h.s.\ of \eqref{int:betaMISE} the first term dominates. For this purpose, we decompose each of the last two terms into two integrals over $|z| \le \vartheta$ and $|z| > \vartheta$, respectively. For all $|z| \le \vartheta$, it holds $|\mathcal{F}(\pi) (z)| \ge 1/2$. Indeed, since $\pi$ is an even density, we have $\mathcal{F}(\pi) (z) = 1 + \int (\exp (\mathrm{i} zy)-1 - \mathrm{i} zy) \pi (y) d y$. Using $|\exp (\mathrm{i} x)-1- \mathrm{i} x| \le |x|^2/2$, $x \in \mathbb{R}$, we get
	
		$|\mathcal{F}(\pi) (z)| \ge 1-  m_2 z^2/2$. By Lemma \ref{mom}, we have 
		$m_2 \le 1/\lambda \le 1/\vartheta^2$.
		On the other hand, the Paley--Wiener theorem implies that $\mathcal{F}(\beta') (z)$ vanishes for $|z| > \vartheta$. Finally, it remains
		\begin{equation}
		\mathbb{E}\left[\int_{\mathbb{R}} | \beta'_{N,T}(y)-\beta' (y) |^2 d y \right]  \lesssim \omega^{-2}  \Big(  \mathbb{E}\left[\int_{\mathbb{R}} | \Psi_{N,T}(y)-\Psi(y) |^2 d y\right]+ 1/N_T \Big) +\vartheta^2/N_T,
		\end{equation}
		where Corollary \ref{cor1} provides an upper bound on the dominating term.


\subsection{Proof of Theorem \ref{minimax}}
We will use the classical two hypotheses approach presented in the monograph \cite{T09}. 
More specifically,
we will find functions $\beta_0,\beta_1 \in \mathcal F_{p, C, C_0, \boldsymbol{a},\lambda}$
such that 
\[
\|\beta'_0-\beta'_1\|_{L^2(\mathbb{R})}^2 = \operatorname{const} \cdot (\log N)^{-p/J} \quad \text{and} \quad
K (\Pi_{\beta_1}^{\otimes N}, \Pi_{\beta_0}^{\otimes N})  \lesssim 1,
\]
where $K (\Pi_{\beta_1}^{\otimes N}, \Pi_{\beta_0}^{\otimes N})$ denotes the 
Kullback-Leibler divergence.
 
We start with some preliminary estimates.
Let $\beta \in \mathcal F_{p, C, C_0, \boldsymbol{a},\lambda}$. Due to the inequality $(\log \pi_\beta)''(y)\leq -2a-(\beta''\star\pi_{\beta})(y) \leq -\lambda$
	the  probability measure $\Pi_\beta$ satisfies the logarithmic Sobolev inequality with constant $2/\lambda$: 
	$$
	\operatorname{Ent}_{\Pi_\beta}(f^2) \le \frac{2}{\lambda} \int_{\mathbb{R}} |f'(y)|^2  \Pi_\beta(dy),
	$$
	where 
	$$
	\operatorname{Ent}_{\Pi_\beta} (f^2) := \int_{\mathbb{R}} f^2(y) \log f^2(y)  \Pi_\beta(dy) - \int_{\mathbb{R}} f^2(y) \Pi_\beta(dy) \log \Big(\int_{\mathbb{R}} f^2(y) \Pi_\beta(dy) \Big), 
	$$
	for every smooth function $f : \mathbb{R} \to \mathbb{R}$ with $\int_{\mathbb{R}} |f'(y)|^2  \Pi_\beta(dy)<\infty$. Hence, for any $\beta_0,\beta_1 \in \mathcal F_{p, C, C_0, \boldsymbol{a},\lambda}$, we can bound the Kullback-Leibler divergence as
	\begin{equation} \label{KLineq}
	K(\Pi_{\beta_1}, \Pi_{\beta_0}) := \int_{\mathbb{R}} \pi_{\beta_1}(y) \log \frac{\pi_{\beta_1}(y)}{\pi_{\beta_0}(y)} dy \leq \frac{1}{2\lambda}  \int_{\mathbb{R}} \pi_{\beta_1}(y)|g(y)|^2 dy
	\end{equation} 
	with a function
	\begin{equation}
	g = \frac{\pi'_{\beta_1}}{\pi_{\beta_1}} - \frac{\pi'_{\beta_0}}{\pi_{\beta_0}}= \Big( \Big( \sum_{1 \le j \le J} a_j \varphi_j  + \beta_0 \Big)' \star \pi_{\beta_0}  \Big) - \Big( \Big( \sum_{1 \le j \le J} a_j \varphi_j  + \beta_1 \Big)' \star \pi_{\beta_1} \Big).
	\end{equation}
	We further decompose it as $g = \sum_{1<j\le J} a_j g_j + g_1 + g_0$ with
	\begin{equation}\label{def:g}
	g_j :=  \varphi'_j \star (\pi_{\beta_0}- \pi_{\beta_1}), \ 1 < j \le J, \qquad g_1 := \beta'_1 \star (\pi_{\beta_0}- \pi_{\beta_1}), \qquad g_0 := (\beta'_{0} - \beta'_{1}) \star \pi_{\beta_0}.
	\end{equation}
	In general, it seems hard to assess the functions $g_j$ for $j\geq 1$ directly (in contrast to $g_0$). Instead, we will show  that
	\begin{equation} \label{usefulin}
	\frac{1}{2\lambda}  \int_{\mathbb{R}} \pi_{\beta_1}(y)|g(y)-g_0(y)|^2 dy \leq \gamma^2 K(\Pi_{\beta_1}, \Pi_{\beta_0})
	\end{equation}
	for some $\gamma\in (0,1)$, and as a consequence of the inequality \eqref{KLineq} we deduce
	that
	\[
	K(\Pi_{\beta_1}, \Pi_{\beta_0}) \lesssim \int_{\mathbb{R}} \pi_{\beta_1}(y)|g_0(y)|^2 dy.
	\]
	The latter bound will be estimated directly for a proper choice of functions $\beta_0,\beta_1$.
	
	We proceed with showing \eqref{usefulin}. We will find a constant $\gamma_j > 0$ such that
	\begin{equation}\label{ineq:intg}
	\frac{1}{2\lambda} \int_{\mathbb{R}} \pi_{\beta_1} (y) |g_j(y)|^2 dy \le \gamma^2_j K(\Pi_{\beta_1},\Pi_{\beta_0}), \qquad1 \le j \le J.
	\end{equation}
	Since
	$\|g_1\|_\infty  \le \| \beta'_1 \|_\infty \|\pi_{\beta_0}-\pi_{\beta_1} \|_{L^1(\mathbb{R})}$ and $\|\pi_{\beta_0}-\pi_{\beta_1} \|_{L^1(\mathbb{R})} \le \sqrt{2 K(\Pi_{\beta_1},\Pi_{\beta_0})}$,
	we have 
	\begin{equation} 
	\gamma^2_1 = \frac{\|\beta'_1\|_\infty^2}{\lambda}.
	\end{equation} 
	For $1 < j \le J$, we have
	\begin{equation} 
	|g_j (y)|^2 \le 

	(2j)^2 \int_{\mathbb{R}} (y-x)^{2(2j-1)} \Big( \sqrt{\pi_{\beta_1}(x)}+\sqrt{\pi_{\beta_0}(x)} \Big)^2 d x \cdot 
	\| \sqrt{\pi_{\beta_1}} - \sqrt{\pi_{\beta_0}} \|_{L^2(\mathbb{R})}^2
	\end{equation}
	where $\|\sqrt{\pi_{\beta_1}} - \sqrt{\pi_{\beta_0}} \|_{L^2(\mathbb{R})}^2 \le K(\Pi_{\beta_1}, \Pi_{\beta_0})$ and 
	$( \sqrt{\pi_{\beta_1}} +\sqrt{\pi_{\beta_0}})^2 \le 2( \pi_{\beta_1} + \pi_{\beta_0})$. Futhermore, 
	\begin{equation}
	\int_{\mathbb{R}} \pi_{\beta_1} (y) \int_{\mathbb{R}} (y-x)^{2k} \pi_{\beta_i} (x) dx dy
	= \sum_{j=0}^{k} \binom{2k}{2j} m_{2j}^{\pi_{\beta_1}} m_{2(k-j)}^{\pi_{\beta_i}} \le \frac{C_k}{\lambda^k},
	\end{equation}
	by Lemma \ref{mom} with $m_j^{\pi_{\beta_i}} = \int_{\mathbb{R}} y^j \pi_{\beta_i} (y) dy$, $i=0,1$, and
	\begin{equation}
	C_k 
	= \sum_{j=0}^k \binom{2k}{2j} (2j - 1)!! (2(k-j) - 1)!!

	= \frac{(2k)!}{k! 2^k} \sum_{j=0}^k \binom{k}{j} = \frac{(2k)!}{k!}.
	\end{equation}
	Hence, \eqref{ineq:intg} holds true with
	\begin{equation}
	\gamma_j^2 = \frac{2 (2j)^2 C_{2j-1}}{\lambda^{2j}}
	\qquad 1 < j \le J.
	\end{equation} 
	We conclude that 
	\begin{align*}
	(K (\Pi_{\beta_1}, \Pi_{\beta_0}))^{1/2} \le \Big(\sum_{1<j \le J} a_j \gamma_j + \gamma_1 \Big) (K (\Pi_{\beta_1}, \Pi_{\beta_0}))^{1/2} +  \Big( \frac{1}{2\lambda}\int_{\mathbb{R}} |g_0 (y)|^2 \pi_{\beta_1}(y) dy\Big)^{1/2}.
	\end{align*}
	Consequently, we deduce the inequality 
	\begin{equation} \label{firstine}
	K (\Pi_{\beta_1}, \Pi_{\beta_0}) \le \frac{1}{2\lambda(1 - \sum_{1<j \le J} a_j \gamma_j - \gamma_1)^2} \int_{\mathbb{R}} |g_0 (y)|^2 \pi_{\beta_1}(y)d y.
	\end{equation}
	Due to \eqref{firstine} we only need to handle the last term in \eqref{def:g}. For this purpose we use following construction: We introduce the constants $\rho>0$, $M>0$ ($\rho\to 0$ and $M\to \infty$ to be chosen later) and a function $\phi\in C^{2}(\mathbb{R}) $ with 
	\[
	\text{supp}(\phi)=[-2,-1] \cup [1,2],
	\]
	and set
	\begin{equation}
	\beta_0(y)=f_0(y), \qquad \beta_1(y)=f_0(y) + \rho M \phi(y/M). 
	\end{equation}
	Here $f_0 \in \mathcal F_{p, C/4, C_0/2, (a_1/2, \dots, a_J/2^J),\lambda/2} \subset \mathcal F_{p, C, C_0, \boldsymbol{a},\lambda}$. 

	To ensure
	that $\beta_1 \in \mathcal F_{p, C, C_0, \boldsymbol{a},\lambda}$ we assume
	\begin{align}\label{condi1}
	\rho M\|\phi\|_{\infty} \le C_0/2, \quad &\rho \|\phi'\|_{\infty} \le C_1- C_1/2^{1/2}, \quad (\rho/M) \|\phi''\|_{\infty} \le C_2/2,\\
	\label{condi2} &\rho^2 M^{2p+1} \int_{\mathbb{R}} y^{2p} |\phi'(y)|^2 dy \le C/4.
	\end{align}





	In particular, condition \eqref{condi2} allows us to choose and later use $\rho = c M^{-p-1/2}$ for some $c>0$.
	Furthermore, the condition $p>1/2$ is required to ensure that  $\rho M=O(1)$.
	We obviously  have that
	\begin{equation}
	\|\beta'_0-\beta'_1\|_{L^2(\mathbb{R})}^2 =\rho^2M \|\phi'\|_{L^2(\mathbb{R})}^2.
	\end{equation}
	Next, we will bound the right hand side of \eqref{firstine}, where recall \begin{equation}
	g_0 (y) = ((\beta'_1-\beta'_0) \star \pi_{\beta_0}) (y).
	\end{equation} 
	For this purpose we note that $g_0(y) = - g_0 (y)$ and then decompose $\int_{\mathbb{R}} =2 (\int_0^k + \int_k^\infty)$, where the threshold $k<M$ will be chosen later. Since $Z_{\pi_{\beta_1}} \geq \int_{\mathbb{R}} \exp(- \sum_{j=0}^J \alpha_j y^{2j} - \|\beta_1\|_{\infty}) dy$ and $\| g_0 \|_\infty \le \| \beta'_0 - \beta'_1 \|_\infty$ we deduce that 
	\begin{equation}
	\int_k^\infty |g_0(y)|^2 \pi_{\beta_1}(y) dy \lesssim \rho^2
	\int_k^\infty \exp(-a_J y^{2J}) dy  \lesssim \rho^2 \frac{\exp(-a_J k^{2J})}{k^{2J-1}}. 
	\end{equation}
	On the other hand, we get 
	\[
	\int_0^k |g_0(y)|^2 \pi_{\beta_1}(y) dy \lesssim \rho^2 k \sup_{y \in [0,k]} | (\phi'(\cdot/M) \star \pi_{\beta_0})(y) |^2
	\] 
	and
	\[
	|(\phi'(\cdot/M) \star \pi_{\beta_0})(y)| \lesssim \int_{-\infty}^{k-M} \pi_{\beta_0}(y) dy \lesssim \frac{\exp(-a_J (M-k)^{2J})}{(M-k)^{2J-1}}.
	\]
	Consequently, choosing $k=M/2$ we deduce from  \eqref{firstine} that 
	\begin{equation}
	K (\Pi_{\beta_1}, \Pi_{\beta_0}) \lesssim \exp(-a_J (M/2)^{2J})
	\end{equation}
	(recall that $\rho = O( M^{-p-1/2})$). Now, choosing $M= 2 ((\log N)/a_J)^{1/(2J)}$ we finally obtain that 
	\[
	K (\Pi_{\beta_1}^{\otimes N}, \Pi_{\beta_0}^{\otimes N}) = N K (\Pi_{\beta_1}, \Pi_{\beta_0}) \lesssim 1.
	\]
	On the other hand, since $\rho= c M^{-p-1/2}$ we deduce that 
	\[
	\|\beta'_0-\beta'_1\|_{L^2(\mathbb{R})}^2 =
	\operatorname{const} \cdot (\log N)^{-p/J},
	\]
	which by \cite[(2.9) and Theorem 2.2(iii)]{T09} completes the proof of our Theorem  \ref{minimax}. 

	

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