text:
\begin{document}

\title
[Limit theorems for
Hermite-driven processes]
{Limit theorems for integral functionals
of Hermite-driven processes}

\author{Valentin Garino}
\address{Universit\'{e} du Luxembourg, D\'epartement de math\'{e}matiques}

\email{valentin.garino@uni.lu}
\thanks{}

\author{Ivan Nourdin}
\address{Universit\'{e} du Luxembourg, D\'epartement de math\'{e}matiques}

\email{ivan.nourdin@uni.lu}
\thanks{}

\author{David Nualart}
\address{Kansas University}

\email{nualart@ku.edu}
\thanks{}

\author{Majid Salamat}
\address{Isfahan university of Technology, Department of Mathematical sciences}

\email{msalamat@iut.ac.ir}
\thanks{}

\subjclass[2000]{60F05, 60G22, 60H05, 60H07.}

\keywords{Hermite processes, chaotic decomposition, fractional Brownian motion (fBm), multiple Wiener-It\^{o} integrals.}

\begin{abstract}
Consider a moving average process $X$ of the form $X(t)=\int_{-\infty}^t x(t-u)dZ_u$, $t\geq 0$, where
$Z$ is a (non Gaussian) Hermite process of order $q\geq 2$ and $x:\mathbb{R}_+\to\mathbb{R}$ is sufficiently integrable.
This paper investigates the fluctuations, as $T\to\infty$, of integral functionals of the form
$t\mapsto \int_0^{Tt }P(X(s))ds$, in the case where $P$ is any given polynomial function.
It extends a study initiated in \cite{Tran}, where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.
\end{abstract}

\maketitle

\allowdisplaybreaks
\section{Introduction}
Hermite processes have become more and more popular in the recent literature, as they occur naturally when we consider limits of partial sums associated with long-range dependent stationary series.
They form a family of stochastic processes, indexed by an integer $q\geq 1$ and a Hurst parameter $H\in(\frac12,1)$,
that contains the fractional Brownian motion ($q=1$) and the Rosenblatt
process ($q=2$) as particular cases.
We refer the reader to Section \ref{sechermite} and the references therein for a precise definition. Of primary importance in the sequel is the
self-similarity  parameter $H_0$, given in terms of $H$ and $q$ by
\begin{equation}\label{hzero}
H_0=1-\frac{1-H}{q}\in(1-\frac1{2q},1).
\end{equation}

The goal of the present paper is to investigate the fluctuations, as $T\to\infty$, of the family of stochastic processes
\begin{equation}\label{intro}
t\mapsto \int_0^{Tt }P(X(s))ds,\quad t\in [0,1]\,\,\mbox{(say)},
\end{equation}
in the case where $P(x)$ is a polynomial function and $X$ is a moving average process of the form
\begin{equation}\label{X}
X(t)=\int_{-\infty}^t x(t-u)dZ_u,\quad t\geq 0,
\end{equation}
with $Z$ a Hermite process  and $x:\mathbb{R}_+\to\mathbb{R}$ a sufficiently integrable function.
We note that integral functionals such as (\ref{intro}) are often encountered in the context of statistical estimation, see e.g.  \cite{Tran} for a concrete example.\\

Let us first consider the case where $q = 1$, that is to say the case where $Z$ is the fractional Brownian motion.
Note that this is the only case where $Z$ is Gaussian, making the study {\it a priori} much simpler and more affordable.
By linearity and passage to the limit, the process $X$ is also Gaussian.
Moreover, it is stationary, since the quantity  $\E[X(t)X(s)]=:\rho(t-s)$ only depends on $t-s$.
For simplicity and without loss of generality, assume that $\rho(0)=1$, that is, $X(t)$ has variance 1 for any $t$.
As is well-known since the eighties (see \cite{BM,DM,Taqqu}), the fluctuations of (\ref{intro})
heavily depends on the centered  Hermite rank of $P$, defined as
the integer $d\geq 1$ such that $P$ decomposes in the form
\begin{equation}\label{hermite}
P =\E[P(X(0))]+ \sum_ {k = d}^\infty a_k H_k,
\end{equation}
with $ H_k $ the $k$th Hermite polynomials and $a_d \neq 0$.
(Note that the sum (\ref{hermite}) is actually finite, since $P$ is a polynomial, so that $\# \{k: \, a_k \neq 0 \} <\infty $.)

The first result of this paper concerns the fractional Brownian motion.
Even if it does not follow directly from the well-known results of Breuer-Major \cite{BM}, Dobrushin-Major \cite{DM} and Taqqu \cite{Taqqu}, the limits obtained are somehow expected.
In particular, the threshold $H = 1-\frac1{2d}$ is well known to specialists.
However, the proof of this result is not straightforward, and requires several estimations  which are interesting in themselves.

\begin{thm}\label{FBM}
Let $Z$ be a fractional Brownian motion of Hurst index $H\in (\frac12, 1)$, and
let $x\in L^1(\R_+)\cap L^{\frac{1}{H}}(\R_+)$.
Consider the moving average process $X$ defined by (\ref{X}) and assume without loss of generality that ${\rm Var}(X(0))=1$
(if not, it suffices to multiply $x$ by a constant).
Finally, let $P(x)=\sum_{n=0}^N a_nx^n$ be a real-valued polynomial function, and let $d\geq 1$ denotes its centered Hermite rank.
\begin{enumerate}
\item If $d\geq 2$ and $H\in(\frac12,1-\frac1{2d})$ then
\begin{equation}\label{cv1}
T^{-\frac12}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
\end{equation}
converges in distribution in $C([0,1])$ to
a standard Brownian motion $W$, up to some multiplicative constant $C_1$ which is explicit and depends only on $x$, $P$ and $H$.
\item If  $H\in(1-\frac1{2d},1)$ then
\begin{equation}\label{cv2}
T^{d(1-H)-1}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
\end{equation}
converges in distribution in $C([0,1])$ to
a Hermite process of index $d$ and Hurst parameter $1-d(1-H)$, up to some multiplicative constant $C_2$ which is explicit and depends only on $x$, $P$ and $H$.
\end{enumerate}
\end{thm}

Now, let us consider the non-Gaussian case, that is, the case where $q\geq 2$.
As we will see, the situation is completely different, both in the results obtained (rather  unexpected) and in the methods used
(very different from the Gaussian case).
Let $L>0$. We define $\mathcal S_L$ to be the set of bounded functions $l:\mathbb{R_+}\rightarrow\mathbb{R}$ such that $y^L l(y)\to 0$ as $y\to\infty$.
We observe that  $\mathcal S_L\subset  L^1(\mathbb{R_+})\cap L^{\frac1H}(\mathbb{R_+})$ for any $L>1$.
We can now state the following result.
\begin{thm}\label{main}
Let $Z$ be a Hermite process of order $q\geq 2$ and Hurst parameter $H\in (\frac{1}{2}, 1)$, and
let $x\in \mathcal S_L$ for some $L>1$.
Recall $H_0$ from (\ref{hzero}) and consider the moving average process $X$ defined by (\ref{X}).
Finally, let $P(x)=\sum_{n=0}^N a_nx^n$ be a real-valued polynomial function. Then, one and only one of the following two situations takes place at $T\to\infty$:
\begin{enumerate}
\item[(i)] If $q$ is odd and if $a_n\neq 0$ for at least one odd $n\in\{1,\ldots,N\}$, then
$$
T^{-H_0}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
$$
converges in distribution in $C([0,1])$ to a fractional Brownian motion of parameter $H_1:=H_0$, up to some multiplicative constant $K_1$ which is explicit and depends only on $x$, $P$, $q$ and $H$, see Remark \ref{rk}.
\item[(ii)] If $q$ is even, or if $q$ is odd and $a_n=0$ for all odd $n\in\{1,\ldots,N\}$, then
$$
T^{1-2H_0}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
$$
converges in distribution in $C([0,1])$ to a Rosenblatt process  of Hurst parameter $H_2:=2H_0-1$, up to some multiplicative  constant $K_2$ which is explicit and depends only on $x$, $P$, $q$ and $H$, see Remark \ref{rk}.
\end{enumerate}
\end{thm}

\begin{rem}\label{rk}
{\rm
Whether in Theorem \ref{FBM} or Theorem \ref{main}, the multiplicative constants appearing in the limit can be all given explicitly by following the respective proofs. For example,
the constant $K_1$ and $K_2$ of Theorem \ref{main} are given by the following intricate expressions:
\begin{eqnarray*}
K_{1}&=&\sum_{n= 3, n\,  {\rm odd}}^Na_{n} c_{H,q}^nK_{x,n,1} \\
K_{2}&=&\sum_{n=2}^N a_{n} c_{H,q}^nK_{x,n,2}+a_1\mathbb{I}_{\{q=2\}}\int_{\R_+} x(\nu) d\nu
\end{eqnarray*}
with
\[
K_{x,n,i}= \sum_{\alpha \in A_{n,q}, nq -2 |\alpha|=i } \frac {C_\alpha  K_{x,\alpha, H_0}} { c_{H_i, i}}, \quad i=1,2,
\]
where the sets and constants in the previous formula are defined in Sections \ref{preli}, \ref{ChaoticExp} and \ref{sec5}.
}
\end{rem}

We note that our Theorem \ref{main} contains as a very particular case the main result of \cite{Tran}, which corresponds to the choice $P(x)=x^2$ and
thus situation (ii). Moreover, let us emphasize that our Theorem \ref{main} not only studies the convergence of finite-dimensional distributions as in \cite{Tran}, but also provides a {\it functional} result.

Because the employed method  is new,
let us sketch the main steps of the proof of Theorem \ref{main}, by using the
classical notation of the Malliavin calculus (see Section 2 for any unexplained definition or result); in particular we write $I_p^B(h)$ to indicate the $p$th multiple Wiener-It\^o integral of kernel $h$ with respect to the standard (two-sided) Brownian motion $B$.

 \medskip

{\bf (Step 1)}  In Section \ref{ChaoticExp}, we represent the moving average process $X$ as a $q$th multiple Wiener-It\^o integral with respect to $B$:
$$
X(t)=c_{H,q}\,I^B_q(g(t,\cdot)),
$$
where  $c_{H,q}$ is an explicit constant and the kernel $g(t,\cdot)$ is given by
\begin{equation}\label{gg}
g(t, \xi_1, \dots, \xi_q)=\int_{-\infty}^tx{(t-v)}\prod_{j=1}^q(v-\xi_j)_+^{H_0-\frac32}dv,
\end{equation}
for $ \xi_1, \dots, \xi_q\in\mathbb{R}$, $ t\geq 0$.

 \medskip

{\bf (Step 2)}
In Lemma \ref{Un} we  compute the chaotic expansion of the $n$th power of $X(t)$ for any $n\geq 2$ and $t>0$, and obtain an expression of the form
$$
X^n(t)=
c^n_{H, q}\sum_{  \alpha \in A_{n,q}}C_\alpha I_{nq-2|\alpha|}^B (\otimes _\alpha ( g(t, \cdot), \dots, g(t, \cdot))),
$$
where we have used the novel notation $ \otimes _\alpha ( g(t, \cdot), \dots, g(t, \cdot))$ to indicate iterated contractions whose precise definition is given in Section \ref{label},
and where $C_\alpha$ are combinatorial constants and the sum runs over a family $A_{n,q}$ of suitable multi-indices $ \alpha
=( \alpha_{ij}, 1\le i<j\le n)$. As an immediate consequence, we deduce that our quantity of interest can be decomposed as follows:
\begin{align}\label{reprintro}
&\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds =a_0\int_0^{Tt} X(s)ds\\
& \hskip1cm+\sum_{n=2}^N a_n \, c_{H,q}^n \notag
\sum_{  \alpha \in A_{n,q}, nq -2 |\alpha| \ge 1}C_\alpha \int_{0}^{Tt} I_{nq-2 |\alpha|}^B (\otimes _\alpha ( g(s, \cdot), \dots, g(s, \cdot)))ds.
\end{align}

 \medskip

{\bf (Step 3)} In Proposition \ref{prop2}, we compute an explicit expression for the iterated contractions  $ \otimes _\alpha ( g(t, \cdot), \dots, g(t, \cdot))$ appearing in the right-hand side of (\ref{reprintro}), by using that $g$ is given by (\ref{gg}).\\

To ease the description of the remaining steps, let us now set
\begin{equation}\label{cf}
F_{n,q,\alpha,T}(t)
=
 \int_0^{Tt} I^B_{nq-2|\alpha|}( \otimes _\alpha ( g(s, \cdot), \dots, g(s, \cdot)))ds.
\end{equation}

 \medskip

{\bf (Step 4)}
As $T\to \infty$, we show in
Proposition  \ref{assympofF}
that, if  $nq-2|\alpha|<\frac{q}{1-H}$, then   $T^{-1+(1-H_0)(nq-2|\alpha|)}F_{n,q,\alpha,T}(t)$  converges in distribution to a Hermite process (whose order and Hurst index are specified) up to some multiplicative constant. Similarly, we prove in Proposition  \ref{prp5} that, if $nq -2|\alpha| \ge 3$, then $T^{\alpha_0}F_{n,q,\alpha,T}(t)$  is tight and converges   in $L^2(\Omega)$ to zero, where $\alpha_0$ is given in (\ref{alpha0}).

  \medskip

{\bf (Step 5)}
By putting together the results obtained in the previous steps, the two convergences stated in Theorem \ref{main} follow immediately.\\

To illustrate a possible use of our results,
we study in Section \ref{sec5bis} an extension of the classical fractional Ornstein-Uhlenbeck process (see, e.g., Cheridito {\it et al} \cite{cheredito2003}) to the case where the driving process is more generally a Hermite process. To the best of our knowledge,
there is very little literature devoted to this mathematical object, only \cite{nourdin2019statistical,slaoui2018limit}. \\

The rest of the paper is organized as follows.
Section \ref{preli} presents some basic results about
multiple Wiener-It\^o integrals and Hermite processes,
as well as some other facts that are
used throughout the paper.
Section \ref{ChaoticExp} contains preliminary results.
The proof of Theorem \ref{FBM} (resp. Theorem \ref{main}) is given in Section \ref{sec4} (resp. Section \ref{sec5}).
In Section \ref{sec5bis}, we provide a complete asymptotic study of the Hermite-Ornstein-Uhlenbeck process, by means of  Theorems \ref{FBM} and  \ref{main} and of an extension of Birkhoff's ergodic Theorem.
Finally,  Section \ref{sec6} contains two technical  results: a power counting theorem and  a version of the Hardy-Littelwood inequality, which both play an important role in the proof of our main theorems.

\section{Preliminaries on multiple Wiener-It\^o integrals and Hermite processes}\label{preli}

\subsection{Multiple Wiener-It\^o integrals and a product formula}\label{label}
A function $f:\mathbb{R}^p\to\mathbb{R}$ is said to be {\it symmetric} if the following relation holds for all permutation $\sigma\in \mathfrak{S}(p)$:
$$
f(t_1, \dots, t_p)=f(t_{\sigma(1)}, \dots, t_{\sigma(p)}),\quad
t_1, \dots, t_p\in\mathbb{R}.
$$
The subset of $L^2(\mathbb{R}^p)$ composed of symmetric functions is denoted by $L_s^2(\mathbb{R}^p)$.

Let $B=\{B(t)\}_{t\in\mathbb{R}}$ be a two-sided Brownian motion. For any given $f\in L^2_s(\mathbb{R}^p)$ we consider the  {\it multiple Wiener-It\^{o} integral} of $f$ with respect to $B$, denoted by
$$
I^B_p(f) =\int_{\mathbb{R}^p}f(t_1, \dots, t_p) dB(t_1)\cdots dB(t_p).
$$
This stochastic integral satisfies $\mathbb{E}[I^B_p(f)]=0$ and
\[
\mathbb{E}[I^B_p(f)I^B_q(g)]=\mathbf{1}_{\{p=q\}} p! \langle f, g\rangle_{L^2(\mathbb{R}^p)}
\]
for $f\in L^2_s(\R^p)$ and $g\in L^2_s(\R^q)$,
see \cite{NP1} and \cite{Nualart} for precise definitions and further details.

It will be convenient in this paper to deal with multiple Wiener-It\^o integrals of possibly nonsymmetric functions.
If $f\in  L^2(\R^p)$, we put  $I_p^B(f)= I_p^B({\tilde f})$, where $\tilde{f}$ denotes the symmetrization of $f$, that is,
\[
\tilde{f}(x_1, \dots, x_p) =\frac 1 {p!} \sum_{\sigma\in  \mathfrak{S}(p)} f(x_{\sigma(1)}, \dots, x_{\sigma(p)}).
\]

 We will need  the expansion as a sum of multiple Wiener-It\^o integrals for  a product of the form $$\prod_{k=1}^n I_q^B(h_k),$$ where $q\ge 2$ is fixed and  the functions $h_k$ belong to $L^2_s(\R^q)$ for $k=1,\dots, n$.  In order to present this extension of the product formula and to  define the relevant contractions between the functions $h_i$ and $h_j$ that will naturally appear, we introduce some further notation.
   Let
$A_{n,q}$ be the set of multi-indices $\alpha= (\alpha_{ij}, 1\le i<j \le n)$ such that, for each $k=1,\dots, n$,
\[
\sum_{1\leq i<j\leq n}  \alpha_{ij} {\bf 1}_{k\in\{i,j\}}    \le q.
\]
Set $|\alpha| = \sum_{1\le i<j \le n} \alpha_{ij}$,
\[
\beta^0_k = q-\sum_{1\leq i<j\leq n}  \alpha_{ij} {\bf 1}_{k\in\{i,j\}}    , \qquad 1\le k\le n
\]
and
\begin{equation} \label{em}
m:= m(\alpha)= \sum_{k=1}^n \beta^0_k  = nq-2|\alpha|.
\end{equation}
For each $ 1\le i < j \le n$, the integer $\alpha_{ij}$ will represent the number of variables in  $h_i$ which are contracted with $h_j$ whereas, for each $k=1,\dots, n$, the integer $\beta^0_k$ is the number of variables in $h_k$ which are not contracted.  We will also write
$\beta_k = \sum_{j=1}^k \beta^0_j$ for $k=1,\dots, n$ and $\beta_0=0$.
Finally, we set
\begin{equation} \label{Calpha}
C_\alpha =\frac {q!^n}{  \prod _{k=1}^n \beta^0_k!  \prod_{ 1\le i  <j\le n}  \alpha_{ij}! }.
\end{equation}
With these  preliminaries,    for any element $\alpha \in A_{n,q}$ we can define the contraction $\otimes_\alpha(h_1, \dots, h_n)$ as the function of $nq-2|\alpha|$ variables obtained by contracting $\alpha_{ij}$ variables between $h_i$ and $h_j$ for each couple of indices $1\le i<j \le n$. Define the collection $(u^{i,j})_{1\leq i,j\leq n,i\neq j}$ in the following way:
$$u^{i,j}=\alpha_{\min(i,j),\max(i,j)}.$$
We then have
\begin{align}   \notag
&\otimes_\alpha(h_1, \dots, h_n)(\xi_1, \dots, \xi_{nq-2|\alpha|}) \\
& =\int_{\mathbb{R}^{|\alpha|}}
 \prod_{k=1}^n h_k( s^{k,1}_1,\dots,s^{k,1}_{u^{k,1}},\dots,s^{k,n}_{1},\dots,s^{k,n}_{u^{k,n}} ,\xi_{1+\beta_{k-1}}, \dots, \xi_{\beta_k} )\\
&\qquad\times\prod_{1\leq i<j\leq n}ds^{i,j}_{1}\dots ds^{i,j}_{u^{i,j}} \notag
\end{align}\label{k4}
Notice that the function  $\otimes_\alpha(h_1, \dots, h_n)$ is not necessarily symmetric.

Then, we have the following result.

\begin{lem}\label{Un}
Let $n,q\geq 2$ be some integers and let $h_i\in L^2_s(\mathbb{R}^q)$ for $i=1,\dots, n$.
We have
\begin{align}\label{Un1}
\prod_{k=1}^n I_q^B(h_k)=\sum_{ \alpha \in A_{n,q}} C_\alpha  I^B_{nq-2|\alpha|}( \otimes _\alpha(h_1, \dots, h
_n) ).
\end{align}
\end{lem}

 \begin{proof}
The product formula for multiple stochastic integrals (see, for instance,  \cite[Theorem 6.1.1]{PT}, or  formula (2.1) in \cite{BN}  for $n=2$)   says that
\begin{equation}  \label{product}
 \prod _{k=1}^n I^B_{q}( h_k)
= \sum_{  \mathcal{P}, \psi }    I^B_{\beta_1^0 + \cdots  +\beta^0_n} \left(   \left( \otimes_{k=1}^n h_k\right)_{\mathcal{P}, \psi} \right),
\end{equation}
where  $\mathcal{P}$ denotes the set of all partitions $\{1,\dots, q\} = J_i \cup \left(\cup_{ k=1,\dots, n, k \not=i} I_{ik}\right) $, where for any  $i,j =1,\dots,  n$, $I_{ij}$ and $I_{ji}$ have the same cardinality $\alpha_{ij}$, $\psi_{ij}$ is a bijection between  $ I_{ij} $
 and $I_{ji}$ and $\beta^0_k = | J_k|$. Moreover,  $ \left( \otimes_{k=1}^n h_k\right)_{\mathcal{P}, \psi}$ denotes the contraction of
 the indexes $\ell$ and $\psi_{ij} (\ell)$ for any  $\ell  \in I_{ij}$ and any  $i,j =1\dots, n$.
Then, formula  (\ref{Un1}) follows form (\ref{product}), by just counting the number of partitions, which is
\[
\prod_{k=1}^n \frac {q!}{ \prod_{i \, {\rm or} \, j \not =k} \alpha_{ij}! \beta^0_k!}
\]
and multiplying by the number of bijections, which is  $\prod_{1\le i< j \le n} \alpha_{ij}!$.
\end{proof}

 Notice that when $n=2$, formula (\ref{Un1}) reduces to the well-known formula for the product of two multiple integrals.  That is,
for any two symmetric functions $f\in L_s^2(\mathbb{R}^p)$ and $g\in L_s^2(\mathbb{R}^q)$  we have
$$
I^B_p(f)I^B_q(g)=\sum_{r=0}^{\min(p, q)}r! \binom{p}{r}\binom{q}{r}I^B_{p+q-2r}(f\otimes_rg).
$$
where, for $0\leq r\leq \min (p, q)$,   $f\otimes_r g\in L^2(\mathbb{R}^{p+q-2r})$ denotes the   contraction  of $r$ coordinates between $f$ and $g$.

\subsection{Hermite processes}\label{sechermite}

Fix $q\geq 1$ and $H\in(\frac12,1)$.
The Hermite process of index $q$ and Hurst parameter $H$  can be  represented by means of a multiple Wiener-It\^{o}
integral with respect to  $B$  as follows, see e.g. \cite{Maej-Tudor}:
\begin{equation}\label{defher}
Z^{H,q}(t) = c_{H, q} \int_{\mathbb{R}^q} \int_{[0,t]}\prod_{j=1}^{q}(s-x_j)^{H_0-\frac32}_+dsdB(x_1)\cdots dB(x_q),\quad t\in\R.
\end{equation}
Here, $x_+=\max\{x, 0\}$, the constant $c_{H, q}$ is chosen to ensure that  \newline ${\rm Var}(Z^{H,q}(1)) = 1$, and
$$H_0=1-\frac{1-H}q\in\big(1-\frac1{2q},1\big).$$

When $q=1$, the  process $Z^{H,1} $ is Gaussian and is nothing but the fractional Brownian motion with Hurst parameter $H$.
For $q\geq 2$, the processes $Z^{H,q} $ are no longer Gaussian: they belong to the $q$th Wiener chaos.
The process $Z^{H,2} $ is known as the Rosenblatt process.

Let $|\mathcal{H}|$ be the following class of functions:
$$
|\mathcal{H}|=\Big\{f:\mathbb{R}\to \mathbb{R}\Big |\,\, \int_\mathbb{R}\int_\mathbb{R}|f(u)||f(v)||u-v|^{2H-2}dudv<\infty\Big\}.
$$
Maejima and Tudor  \cite{Maej-Tudor} proved that the stochastic integral $\int_\mathbb{R} f(u)dZ^{H,q}(u)$ with respect to the Hermite process $Z^{H,q}$ is well defined when $f$ belongs to $|\mathcal{H}|$.
Moreover, for any order $q\geq 1$, index $H\in(\frac12, 1)$
and function $f\in|\mathcal{H}|$,
\begin{eqnarray}\label{integral}
&&\int_\mathbb{R} f(u)dZ^{H,q}(u)\\
&=&c_{H, q} \int_{\mathbb{R}^q}\left(\int_{\mathbb{R}}f(u)\prod_{j=1}^{q}(u-\xi_j)_+^{H_0-\frac 32 }du\right)dB(\xi_1) \cdots dB(\xi_q).\notag
\end{eqnarray}
As a consequence of the Hardy-Littlewood-Sobolev inequality featured in \cite{Beckner},
we observe that $L^1(\mathbb{R})\cap L^{\frac1H}(\mathbb{R})\subset
|\mathcal{H}|$.

\section{Chaotic decomposition of $\int_0^{Tt }P(X(s))ds$}\label{ChaoticExp}
Assume $x\in|\mathcal{H}|$ and $q\geq 1$. Using \eqref{integral} and bearing in mind the notation and results from Section \ref{preli}, it is immediate that $X$ can be written as
\begin{equation}\label{Xbis}
X(t)=c_{H, q}I^B_q(g(t, .)),
\end{equation}
where $g(t, .)$ is given by
\begin{equation}\label{g-T}
g(t, \xi_1, \dots, \xi_q)=\int_{-\infty}^tx{(t-v)}\prod_{j=1}^q(v-\xi_j)_+^{H_0-3/2}dv,
\end{equation}
and $c_{H, q}$ is defined as in (\ref{defher}).

\subsection{Computing the chaotic expansion of $X(t)^n$ when $n\geq 2$}

 Let us denote by $A_{n,q}^0$ the set of elements $\alpha \in  A_{n,q}$ such that $nq-2|\alpha|=0$ and $A_{n,q}^1$ will be  the set of elements $\alpha \in  A_{n,q}$ such that $nq-2|\alpha|\ge 1$.
Notice that   when $nq$ is odd, $A_{n,q}^0$ is empty.
Using \eqref{Un1}, we obtain the following formula for the expectation of the $n$th power ($n\geq 2$) of $X$ given by \eqref{X}:
\begin{equation}\label{expec}
   \mathbb{E}[X(t)^n]=  (c_{H, q})^n\sum_{  \alpha \in A_{n,q}^0}C_\alpha I_{nq-2|\alpha|}^B (\otimes _\alpha ( g(t, \cdot), \dots, g(t, \cdot))).
   \end{equation}
We observe in particular that $\mathbb{E}[X(t)^n]=0$ whenever $nq$ is odd.
From \eqref{Un1} and \eqref{expec}, we deduce for $n\geq 2$ that

\begin{align*}
X(t)^n - \mathbb{E}[X(t)^n]=(c_{H, q})^n\sum_{  \alpha \in A_{n,q}^1}C_\alpha I_{nq-2|\alpha|}^B (\otimes _\alpha ( g(t, \cdot), \dots, g(t, \cdot))),
\end{align*}
implying in turn
\begin{align}\label{repr}
&\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds =
a_1\int_0^{Tt} X(s)ds \\
& \qquad \qquad  + \sum_{n=2}^N a_n
(c_{H, q})^n\sum_{  \alpha \in A_{n,q}^1}C_\alpha  \int_0^{Tt} I_{nq-2|\alpha|}^B (\otimes _\alpha ( g(s, \cdot), \dots, g(s, \cdot)))ds.
\notag
\end{align}

\subsection{Expressing the iterated   contractions of $g$}
We now compute an explicit expression for the iterated   contractions appearing in (\ref{repr}).

\begin{prp}\label{prop2}
Fix  $n\geq 2$, $q\geq 1$ and $\alpha\in A_{n,q}$.
We have
\begin{align*}
&\otimes _\alpha (g(t, .) , \dots, g(t, .))(\xi)
= \beta(H_0-\frac12,2-2H_0)^{|\alpha |}     \\
&  \qquad \times  \int_{(-\infty,t]^n}dv_1\ldots dv_n\prod_{k=1}^n x(t-v_k)
\prod_{1\leq i<j\leq n}|v_i-v_j|^{(2H_0-2)\alpha_{ij}}  \\
& \qquad   \times \prod_{k=1}^{n}\prod_{\ell=1+\beta_{k-1}}^{\beta_k} (v_k - \xi_{\ell})_+^{H_0-\frac32},
\end{align*}
with the convention $\beta_0=0$.
\end{prp}
\begin{proof}
The proof is a straightforward consequence of the following identity
\begin{equation}\label{bienutile}
\int_{\mathbb{R}}(v-\xi)_+^{H_0-3/2}(w-\xi)_+^{H_0-3/2} d\xi=\beta(H_0-\frac12, 2-2H_0)|v-w|^{(2H_0-2)},
\end{equation}
whose proof is elementary, see e.g. \cite{Biagini}.
\end{proof}

\section{Proof of Theorem \ref{main}}\label{sec5}

We are now ready to prove Theorem \ref{main}.
To do so, we will mostly rely on the forthcoming Proposition \ref{assympofF}, which might be a result of independent interest by itself, and
which studies the asymptotic behavior of $F^B_{n,q,\alpha, T}$ given by (\ref{cf}).
We will denote by f.d.d. \!the convergence in law of the finite-dimensional distributions of a given process. Notice that the hypothesis on $x$ is a bit weaker than the one in the main theorem, the fact that $x\in \mathcal{S}_L$ being required in the forthcoming Proposition \ref{prp5}.

\begin{prp}\label{assympofF}
Fix $n\geq 2$, $q\geq 1$ and $\alpha\in A_{n,q}$. Assume the function $x$ belongs to $L^1(\mathbb{R}_+)\cap L^{\frac1H}(\mathbb{R}_+)$, recall $H_0$ from (\ref {hzero})  and let $m$  be defined as  in  (\ref{em}).  Finally, assume that $2m< \frac{q}{1-H}$ (which is automatically satisfied when $m=1$ or $m=2$). Then, as $T\to\infty$,
\begin{equation} \label{fdd}
(T^{-1+(1-H_0)m}F_{n,q,\alpha, T} (t))_{t\in [0,1]} \stackrel{f.d.d.}{\longrightarrow}  \left( \frac {C_\alpha K_{x,\alpha,H_0}} { c_{H(m),m}} Z^{H(m),m}(t) \right)_{t\in [0,1]},
\end{equation}
where  $Z^{H(m),m}$ denotes the
 $m$th Hermite process  of Hurst index $H(m)= 1-\frac mq(1-H)$ and the constants $C_\alpha$ and $ K_{x,\alpha,H_0}$ are defined in (\ref{Calpha}) and (\ref{Kalpha}), respectively.
 Furthermore, $\{T^{-1+(1-H_0)m}(F_{n,q,\alpha, T}(t))_{t\in [0,1]},\,T>0\}$ is tight in $C([0,1])$.
\end{prp}

\begin{proof}
Let $n\geq 2$, $q\ge 1$ and $\alpha \in A_{n,q}$.

\medskip
\noindent
\underline{\it Step 1}:  We will first show the convergence  (\ref{fdd}).
We will   make several change of variables in order to transform the expression of $F_{n,q,\alpha, T}(t)$.  By means of an application of stochastic's Fubini's theorem, we can write
\[
F_{n,q,\alpha, T}(t) = C_\alpha \int_{\R^m}    \Psi_T(\xi_1, \dots, \xi_m) dB(\xi_1) \cdots dB(\xi_m),
\]
where
\begin{align*}
\Psi_T(\xi_1, \dots, \xi_m) &:= T^{-1+m(1-H_0)} \int_0^{Tt} ds \int_{(-\infty, s]^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(s-v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (v_k -\xi_\ell)_+^{H_0 -\frac 32}.
\end{align*}
Using the change of variables $s\to Ts$ and $v_k \to Ts-v_k$, $1\le k\le n$, we obtain
\begin{align*}
\Psi_T(\xi_1, \dots, \xi_m) &:= T^{m(1-H_0)} \int_0^{t} ds \int_{(-\infty, Ts]^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(Ts-v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (v_k -\xi_\ell)_+^{H_0 -\frac 32} \\
&=  T^{-\frac m2} \int_0^{t} ds \int_{[0,\infty)^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (s-\frac{v_k}T - \frac{\xi_\ell}T)_+^{H_0 -\frac 32}.
\end{align*}
By the scaling property of the Brownian motion, the processes
\[
(F_{n,q,\alpha, T}(t))_{ \alpha \in A_{n,q}, 2\le n\le N, t\in [0,1]}
\]
and
\[
(\widehat{F}_{n,q,\alpha, T}(t))_{ \alpha \in A_{n,q}, 2\le n\le N, t\in [0,1]}
\]
have the same  probability distribution, where
\[
\widehat{F}_{n,q,\alpha, T}(t) = C_\alpha \int_{\R^m}    \widehat{\Psi}_T(\xi_1, \dots, \xi_m)dB(\xi_1) \cdots dB(\xi_m)
\]
\begin{align*}
\widehat{\Psi}_T(\xi_1, \dots, \xi_m) &:=     \int_0^{t} ds \int_{(-\infty, 0]^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (s-\frac{v_k}T - \xi_\ell)_+^{H_0 -\frac 32}.
\end{align*}
Set
\[
\widehat{\Psi}(\xi_1, \dots, \xi_m) :=   K_{x,\alpha, H_0}   \int_0^{t} ds    \prod_{\ell = 1 }^m (s  - \xi_\ell)_+^{H_0 -\frac 32},
\]
where
\begin{equation} \label{Kalpha}
K_{x,\alpha, H_0} = \int_{\R_+^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(v_k) \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }.
\end{equation}
Notice that, by Lemma \ref{HLS-lemme}, $K_{x,\alpha, H_0} $ is well defined.
We claim that
\begin{equation} \label{k1}
\lim_{T\rightarrow \infty}  \widehat{\Psi}_T =\widehat{\Psi},
\end{equation}
where the convergence holds in $L^2(\R^m)$. This will imply the convergence in $L^2(\Omega)$ of
$\widehat{F}_{n,q,\alpha, T}(t)$, as $T\rightarrow \infty$ to a Hermite process of order $m$, multiplied by the constant $C_\alpha K_{x,\alpha,H_0}$.

\medskip
\noindent
{\it Proof of (\ref{k1}):}  It suffices to show that the inner products $\langle \widehat{\Psi}_T, \widehat{\Psi}_T \rangle_{L^2(\R^m)}$
and $\langle \widehat{\Psi}_T, \widehat{\Psi} \rangle_{L^2(\R^m)}$ converge, as $T\rightarrow \infty$,  to
\[
\| \widehat{\Psi} \| ^2_{L^2(\R^m)}= K_{x,\alpha, H_0}^2  \beta(H_0 -\frac 12, 2-2H_0)^m  \int_{[0,t]^2} ds ds' |s-s'| ^{(2H_0-2)m},
\]
which is finite because $m<\frac 1{2(1-H_0)} = \frac q{2(1-H)}$. We will show the convergence of $\langle \widehat{\Psi}_T, \widehat{\Psi}_T \rangle_{L^2(\R^m)}$
and the second term can be handled by the same arguments.
We have
\begin{align*}
\| \widehat{\Psi}_T \|_{L^2(\R^m)}^2&=
\int_{[0,t]^2} ds ds' \int_{\R_+^{2n}} dv_1 \cdots dv_n  dv'_1 \cdots dv'_n   \\
&\times \prod_{k=1} ^n x(v_k) x(v'_k)
  \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} } | v'_i-v'_j|^{(2H_0-2)\alpha_{ij} } \\
 &\times \prod_{k=1}^n   \beta(H_0 -\frac 12, 2-2H_0)^{\beta_k} |s-s' -\frac{v_k-v'_k}T |^{(2H_0 -2)\beta_k}.
\end{align*}
Let us first show that given $w_k \in \R$, $1\le k\le n$,
\begin{equation} \label{k2}
\lim_{T\rightarrow \infty} \int_{[0,t]^2} ds ds'  \prod_{k=1}^n  |s-s' -\frac{w_k}T |^{(2H_0 -2)\beta_k} =\int_{[0,t]^2} ds ds'    |s-s'   |^{(2H_0 -2)m}
\end{equation}
and, moreover,
\begin{equation} \label{k3}
\sup_{w_k\in\R, 1\le k\le n}   \int_{[0,t]^2} ds ds'  \prod_{k=1}^n  |s-s' -{w_k} |^{(2H_0 -2)\beta_k}< \infty.
\end{equation}
By the dominated convergence theorem and using  Lemma \ref{HLS-lemme}, (\ref{k2}) and (\ref{k3}) imply (\ref{k1}).

To show (\ref{k2}), choose $\epsilon$ such that  $ |w_k| /T <\epsilon$, $1\le k\le n$,  for $T$ large enough (depending on the fixed $w_k$'s). Then, we can write
\begin{align*}
& \int_{[0,t]^2} ds ds'  \left|  \prod_{k=1}^n  |s-s' -\frac{w_k}T |^{(2H_0 -2)\beta_k} -    |s-s'   |^{(2H_0 -2)m}  \right| \\
& \quad  \le t \int_{ |\xi| >2\epsilon }  d\xi \left|  \prod_{k=1}^n  |\xi -\frac{w_k}T |^{(2H_0 -2)\beta_k} -    |\xi   |^{(2H_0 -2)m}  \right| \\
& \qquad + 2 t  \sup_{|w_k| <\epsilon}  \int_{ |\xi| \le 2\epsilon }  d\xi  \prod_{k=1}^n  |\xi -w_k |^{(2H_0 -2)\beta_k} \\
& \quad := B_1 + B_2.
\end{align*}
The term $B_1$ tends to zero a $T\rightarrow \infty$, for each $\epsilon>0$.  On the other hand, the term $B_2$ tends to zero as $\epsilon \rightarrow 0$. Indeed,
\[
B_2=
2 t   \epsilon ^{(2H_0-2)m +1}  \sup_{|w_k| <1}  \int_{ |\xi| \le 2 }   d\xi  \prod_{k=1}^n  |\xi -w_k |^{(2H_0 -2)\beta_k}.
\]
Note that the above supremum is finite because the function $(w_1, \dots, w_k) \to \int_{ |\xi| \le 2 }   d\xi  \prod_{k=1}^n  |\xi -w_k |^{(2H_0 -2)\beta_k}$ is continuous.

Property (\ref{k3}) follows immediately from the fact that the function
\[
(w_1, \dots, w_k) \to  \int_{[0,t]^2} ds ds'  \prod_{k=1}^n  |s-s' -{w_k} |^{(2H_0 -2)\beta_k}
\]
is continuous and vanishes as $|(w_1, \dots, w_k)|$ tends to infinity.

 We have $H_0=1-\frac{1-H}{q}=1-\frac{1-H(m)}{m}$ with $H(m)$ as above. As a result, we obtain
 the  convergence of the finite-dimensional distributions of  $T^{-1+(1-H_0)m}F_{n,q,\alpha, T}(t)$ to those the $m$th Hermite process
$ Z^{H(m),m}$ multiplied by the constant $\frac {C_\alpha K_{x,\alpha,H_0}} { c_{H(m),m}} $.

\bigskip
\noindent
\underline{\it Step 2}: {\it Tightness}.  Fix $0\le s<t \le 1$. To check that tightness holds in $C([0,1])$, let us  compute  the squared $L^2(\Omega)$-norm
\[
\Phi_T :=T^{-1+(1-H_0)m} \E(| F_{n,q,\alpha, T}(t) -F_{n,q,\alpha, T}(s)|^2).
\]
Proceeding as in the first step of the proof, we obtain
\begin{align*}
\Psi_T &=\E \Bigg(  \Bigg| \int_{\R^m} dB(\xi_1) \cdots dB(\xi_m) \int_s^{t} du \int_{\R_+^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (u-\frac{v_k}T - \xi_\ell)_+^{H_0 -\frac 32} \Bigg| ^2 \Bigg) \\
& \le m!   \int_{\R^m} d\xi_1 \cdots d\xi_m \Bigg| \int_s^{t} du \int_{\R_+^n} dv_1 \cdots dv_n \prod_{k=1} ^n x(v_k) \\
&\times \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} }
\prod_{k=1}^n \prod_{\ell = 1+ \beta_{k-1}}^{\beta_k} (u-\frac{v_k}T - \xi_\ell)_+^{H_0 -\frac 32} \Bigg| ^2.
\end{align*}
Using  (\ref{bienutile}) yields
\begin{align*}
\Psi_T & \le m!
\int_{[s,t]^2} du du' \int_{\R_+^{2n}} dv_1 \cdots dv_n  dv'_1 \cdots dv'_n   \\
&\times \prod_{k=1} ^n x(v_k) x(v'_k)
  \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} } | v'_i-v'_j|^{(2H_0-2)\alpha_{ij} } \\
 &\times \prod_{k=1}^n   \beta(H_0 -\frac 12, 2-2H_0)^{\beta_k} |u-u' -\frac{v_k-v'_k}T |^{(2H_0 -2)\beta_k} \\
  & \le m! (t-s)
\int_{-1}^1 d\xi \int_{\R_+^{2n}} dv_1 \cdots dv_n  dv'_1 \cdots dv'_n   \\
&\times \prod_{k=1} ^n x(v_k) x(v'_k)
  \prod_{1\le i<j \le n} | v_i-v_j|^{(2H_0-2)\alpha_{ij} } | v'_i-v'_j|^{(2H_0-2)\alpha_{ij} } \\
 &\times \prod_{k=1}^n   \beta(H_0 -\frac 12, 2-2H_0)^{\beta_k} |\xi-\frac{v_k-v'_k}T |^{(2H_0 -2)\beta_k} \\
 & \le C (t-s).
\end{align*}
Then the equivalence of all $L^p(\Omega)$-norms, $p\ge 2$,  on a fixed Wiener chaos, also known as the hypercontractivity property, allows us to conclude the proof of the tightness.
 \end{proof}

We will make use of  the notation
\begin{equation} \label{alpha0}
\alpha_0=(1-2H_0){\bf 1}_{\{\mbox{$nq$ is even}\}}-H_0{\bf 1}_{\{\mbox{$nq$ is odd}\}}.
\end{equation}

\begin{prp}\label{prp5}
Fix $n,q\geq 2$ and $\alpha\in A_{n,q}$, assume that the function $x$ belongs to $\mathcal S_L$ for some $L>1$ and that $ m\geq 3$. Then  for any $t\in [0,1]$,
  $ T^{\alpha_0}F_{n,q,\alpha, T}(t)$ converge in $L^2(\Omega)$  to zero as $T\rightarrow \infty$; furthermore, the family
$\{ (F_{n,q,\alpha, T}(t))_{t\in [0,1]} , T>0\} $ is tight in $C([0,1])$.
\end{prp}

\begin{proof}
If $  (2H_0-2)m >-1$, by Proposition \ref{assympofF}, we know that
$T^{-1+m(1-H_0)}  F_{n,q,\alpha, T}(t)$ converges to zero in $L^2(\Omega)$ as $T\rightarrow \infty$.
This implies the convergence to zero   in $L^2(\Omega)$ as $T\rightarrow \infty$ of  $ T^{-\alpha_0}F_{n,q,\alpha, T}(t)$  because
$-1+m(1-H_0) > \alpha_0$.
 We should then concentrate on the case $  (2H_0-2)m \leq-1$.
Once again, we shall divide the proof in two steps:

\bigskip
\noindent
 \underline{\textit{Step 1}}: Let us first prove the convergence in $L^2(\Omega)$.
Fix $\alpha\in A_{n,q}$. We are going to show that
\[
 \lim_{T\rightarrow \infty} T^{2\alpha_0} \E \left( | F_{n,q,\alpha, T} (t) |^2 \right) =0.
 \]
 We know that
\[
 T^{2\alpha_0} \E \left( | F_{n,q,\alpha, T} (t) |^2 \right) = T^{2\alpha_0}  m!
 \times  \left\|  \int_{[0,Tt]} ds \otimes_\alpha (
g(s,\cdot) , \dots, g(s, \cdot))  \right\|^2_{L^2(\mathbb{R}^{ m})}.
\]
In view of the expression  for the contractions obtained in Proposition \ref{assympofF}, it suffices to show that
  \begin{align*}
 & \lim_{T\rightarrow \infty} T^{2\alpha_0}   \int_{[0,Tt]^2} ds ds'  \int_{\mathbb{R}^{ m}}   \int_{(-\infty,s]^n} \int_{(-\infty,s']^n}
  dv_1 \cdots dv_n dv'_1 \cdots dv'_n d\xi_1 \cdots d\xi_{m}\\
  &\times \prod_{k=1}^n x(s-v_k) x(s'-v'_k) \prod_{1 \le i<j\le n} |v_i-v_j|^{(2H_0-2)\alpha_{ij}}|v'_i-v'_j|^{(2H_0-2)\alpha_{ij}} \\
  &\times \prod_{k=1}^n  \prod  _{\ell=1+  \beta_{j-1}} ^{ \beta_j} (v_k -\xi_{\ell})_+^{H_0-\frac 32}  \prod  _{\ell=1+   \beta_{j-1}} ^{ \beta_j} (v'_k -\xi_{\ell})_+^{H_0-\frac 32}=0.
  \end{align*}
  Integrating  in the variables $\xi$'s  and using \eqref{bienutile}, it remains to show that
   \begin{align*}
 & \lim_{T\rightarrow \infty} T^{2\alpha_0}   \int_{[0,Tt]^2} ds ds'      \int_{(-\infty,s]^n} \int_{(-\infty,s']^n}
  dv_1 \cdots dv_n dv'_1 \cdots dv'_n \\
  &\times  \prod_{k=1}^n x(s-v_k) x(s'-v'_k)  \prod_{1 \le i<j\le n} |v_i-v_j|^{(2H_0-2)\alpha_{ij}}|v'_i-v'_j|^{(2H_0-2)\alpha_{ij}}   \\ &\times \prod_{k=1}^n   |v_k-v'_k|^{(2H_0-2)\beta_k}=0.
  \end{align*}
  Set
  \begin{align*}
 \Phi_T & :=  T^{2\alpha_0}   \int_{[0,Tt]^2} ds ds'      \int_{(-\infty,s]^n} \int_{(-\infty,s']^n}
  dv_1 \cdots dv_n dv'_1 \cdots dv'_n \\
  &\times   \prod_{k=1}^n x(s-v_k) x(s'-v'_k)  \prod_{1 \le i<j\le n} |v_i-v_j|^{(2H_0-2)
  \alpha_{ij}}|v'_i-v'_j|^{(2H_0-2)\alpha_{ij}} \\
  &\times \prod_{k=1}^n   |v_k-v'_k|^{(2H_0-2)\beta_k}.
  \end{align*}
  Making the change of variables $w_k=s-v_k$, $w'_k=s'-v'_k$ for $k=1,\dots,n$, yields
    \begin{align*}
 \Phi_T & =  T^{2\alpha_0}   \int_{[0,Tt]^2} ds ds'      \int_{\R_+^n} \int_{[0,\infty)^n}
  dw_1 \cdots dw_n dw'_1 \cdots dw'_n \\
  &\times  \prod_{k=1}^n x(w_k) x(w'_k)  \prod_{1 \le i<j\le n} |w_i-w_j|^{(2H_0-2)\alpha_{ij}}
  |w'_i-w'_j|^{(2H_0-2)\alpha_{ij}}  \\
  &\times\prod_{k=1}^n   |s-s'-w_k+w'_k|^{(2H_0-2)\beta_k}.
  \end{align*}
  Now we use Fubini's theorem and  make  the change of variables $s-s'=\xi$   to obtain
      \begin{align*}
 \Phi_T &  =  tT^{2\alpha_0+1}   \int_{\R_+^{2n}}
  dw_1 \cdots dw_n dw'_1 \cdots dw'_n \\
  &\times  \prod_{k=1}^n |x(w_k) x(w'_k) | \prod_{1 \le i<j\le n} |w_i-w_j|^{(2H_0-2)\alpha_{ij}}
  |w'_i-w'_j|^{(2H_0-2)\alpha_{ij}}  \\
  &\times  \int_{-tT }^{tT}  d\xi  \prod_{k=1}^n   |\xi-w_k+w'_k|^{(2H_0-2)\beta_k}.
  \end{align*}

   We shall distinguish again two subcases:

\bigskip

  \noindent
     {\it Case $  (2H_0-2)m < -1$}:
     Notice that the exponent $2\alpha_0+1$ is negative:
     \begin{itemize}
 \item[(i)] If $nq$ is even, then $\alpha _0 =1-2H_0$ and
 \[
 2\alpha_0+1 =3-4H_0 <0
 \]
 because $H_0 >\frac 34$.
  \item[(ii)]  If $nq$ is odd, then $\alpha _0 =-H_0$ and
 \[
 2\alpha_0+1=1-2H_0<0.
 \]
     \end{itemize}
     Therefore,  in order to show that $ \lim_{T\rightarrow \infty} \Phi_T =0$, it suffices to check that
   \begin{align}
  &  J:= \int_{\R^{2n}}         \notag
  dw_1 \cdots dw_n dw'_1 \cdots dw'_n   \prod_{k=1}^n |x(w_k) x(w'_k)|  \\  \notag
  & \qquad  \times  \prod_{1 \le i<j\le n} |w_i-w_j|^{(2H_0-2)\alpha _{ij}}
  |w'_i-w'_j|^{(2H_0-2)\alpha_{ij}}  \\  \label{ecu1}
  & \qquad \times  \int_{\mathbb{R}}  d\xi  \prod_{k=1}^n   |\xi -w_k+w'_k|^{(2H_0-2)\beta_k} <\infty,
  \end{align}
  where, by convention $x(w)=0$ if $w<0$.
We will apply the Power Counting Theorem  \ref{powercounting} to prove that this integral is finite.
We consider functions on $\R^{2n+1}$ with variables
 $\{(w_k)_{k\leq n},(w'_k)_{k\leq n},\xi\}$.
 The set of linear functions is
\begin{align*}
T&= \{\omega_k,  \omega'_k, 1\le  k\leq n\} \cup\{ w_i- w_j, w'_i-w'_j , 1\leq i<j\leq n\}\\
&\cup\{\xi-w_k+w'_k, 1\leq k\leq n\}.
\end{align*}
The corresponding exponents $(\mu_M,\nu_M)$ for each
  $M\in T$ are $(0,-L)$ for the  linear  functions $w_k$ and $w'_k$ (taking into account that $x\in \mathcal{S}_L$),
  $ (2H_0-2) \alpha_{ij}$ for each function of the form $w_i- w_j$ or $ w'_i-w'_j$  and $(2H_0-2)\beta_k$ for each function of the form
 $ \xi-w_k+w'_k$.

  Then  $J<\infty$, provided conditions (a) and (b)  are satisfied.

\medskip
\noindent
$\bullet$ \textit{Verification of (b)}: Let $W\subset T$ be a linearly independent proper subset of $T$, and
\[
 d_\infty=2n+1-{\rm dim} ({\rm Span}(W))+\sum_{M\in T\setminus ({\rm Span}(W)\cap T)}\nu_M.
 \]
Let $S$ be the following subset of $T$: $S=\{ w_k, w'_k, 1\le k\le n\}$. Let $e={\rm Card}(S\cap {\rm Span}(W))$.
Consider the following two cases:
\begin{itemize}
  \item[(i)]  There exists $  k\leq n$ such that $\xi-w_k+w_k'\in {\rm Span}(W)\cap T$. Then  ${\rm dim}({\rm Span}(W))\geq e+1$. As a consequence,
\begin{align*}
d_{\infty}\leq 2n+1-(e+1)-(2n-e)L<0,
\end{align*}
because $L>1$ and in this case, we should have $e<2n$ because $W$ is a proper subset of $T$.
\item[(ii)] Otherwise,
\begin{align*}
d_{\infty}\leq 2n+1-e-(2n -e)L+(2H_0-2)m<0,
\end{align*}
because $L>1$ and $(2H_0-2)m<-1$.
\end{itemize}

$\bullet$ \textit{Verification of (a)}: A direct verification would require to solve a seemingly difficult combinatorial problem. We can simply remark that
\medskip
\begin{align*}
  &  \int_{[-1,1]^{2n}}
  dw_1 \cdots dw_n dw'_1 \cdots dw'_n   \\
  & \qquad  \times  \prod_{1 \le i<j\le n} |w_i-w_j|^{(2H_0-2)\alpha _{ij}}
  |w'_i-w'_j|^{(2H_0-2)\alpha_{ij}}  \\
  &\qquad\times  \int_{-1}^1  d\xi  \prod_{k=1}^n   |\xi -w_k+w'_k|^{(2H_0-2)\beta_k}\\
 =&m!\frac{1}{\beta(H_0-\frac12,2-2H_0)^{|\alpha |} }\mathbb{E}\left[\left(\int_0^1I^B_{nq-2|\alpha|}(f(s,\cdot),\ldots,f(s,\cdot))\right)^2\right]<\infty
\end{align*}
where $f(s,\xi_1,\ldots \xi_q)=\int_{-\infty}^{+\infty}\mathbb{I}_{[-1,1]}(s-v)\prod_{j=1}^q(v-\xi_j)_+^{H_0-3/2}dv$. Since
$x\in\mathcal{S}_L$, $x$ is bounded on $[-1,1]$. This implies that (a) is verified by the converse side of the Power Counting Theorem.

\medskip
\noindent
     {\it Case $  (2H_0-2)m =-1$}:
     In this case, we can apply  H\"older and Jensen inequalities to $\Phi_T$ in order to get
\begin{align*}
\Phi_T\leq T^{2\alpha_0+1 }A^\frac{\epsilon}{1+\epsilon}B^\frac{1}{1+\epsilon},
\end{align*}
with $2\alpha_0+1<0$,  $A=\left( \int_{\R} |x(w)| dw \right)^{2n}$ and
\begin{align*}
 B&=\int_{\mathbb{R}^{2n}}
  dw_1 \cdots dw_n dw'_1 \cdots dw'_n   \prod_{k=1}^n |x(w_k) x(w'_k)|  \\  \notag
  & \qquad  \times  \prod_{1 \le i<j\le n} |w_i-w_j|^{(2H'_0-2)\alpha_{ij}}
  |w'_i-w'_j|^{(2H'_0-2)\alpha_{ij}}  \\  \label{ecu1}
  & \qquad \times  \int_{\mathbb{R}}  d\xi  \prod_{k=1}^n   |\xi -w_k+w'_k|^{(2H'_0-2)\beta_k},
\end{align*}
where $H'_0= H_0( 1+\epsilon) -\epsilon$. If $\epsilon$ is small enough, $H'_0$   can still be expressed as $1-\frac{1-H'}{q}$ for some $\frac{1}{2}<H'<H$. Moreover, in this case $(2H'_0-2)m<-1$ so we are exactly in the situation of the previous case, and the integral $B$ is finite.

\medskip
\noindent
  \underline{\textit{Step 2}}:
Using the same arguments as previously and the hypercontractivity property, we deduce that there exists a constant $K>0$ such that
for all $0\le s<t \le 1$,
\begin{equation*}
 \mathbb{E}\left( | F_{n,q,\alpha, T} (t)-F_{n,q,\alpha, T} (s) |^4 \right)\leq K|t-s|^2,
\end{equation*}
which proves the tightness in $C([0,1])$.

\end{proof}

It remains to study what happens when $n=1$.
The proof of Proposition \ref{n=1} is very similar to
that of Proposition \ref{assympofF} (although much simpler)   and details are left to the reader.

\begin{prp}\label{n=1}
Fix $q\geq 1$ and assume the function $x$ belongs to $L^1(\mathbb{R}_+)\cap L^{\frac1H}(\mathbb{R}_+)$.
Then   the finite-dimensional distributions of the process
\begin{equation}\label{paris}
G_T(t):=   T^{q(1-H_0)-1} \int_0^{Tt} ds I^B_q (g(s,\cdot)), \quad t\in [0,1]
\end{equation}
  converge in law  to those of a  $q$th Hermite process of Hurst parameter $1-q(1-H_0)$ multiplied by the constant
  $c_{H_0,q}^{-1} \int_0^\infty x(w) dw$,
and the family   \newline $\{(G_T(t))_{t\in [0,1]},\,T>0\}$ is tight in $C([0,1])$.
\end{prp}

We are now ready to make the proof of Theorem \ref{main}.

\medskip

\noindent
{\it Proof of Theorem \ref{main}}. It suffices to consider the decomposition
(\ref{repr}) and to apply the results shown in Propositions  \ref{assympofF} and   \ref{prp5}.
\qed

\section{Proof of Theorem \ref{FBM}}\label{sec4}

Let $Z$ be a fractional Brownian motion of Hurst index $H\in (\frac12, 1)$, and
let $x\in L^1(\R_+)\cap L^{\frac{1}{H}}(\R_+)$.
Consider the moving average process $X$ defined by
$$
X(t)=\int_{-\infty}^t x(t-u)dZ_u,\quad t\geq 0,
$$
which is easily checked to be a stationary centered Gaussian process.
Denote by $\rho:\R\to\R$ the correlation function of $X$, that is, $\rho(t-s)=\E[X(t)X(s)]$, $s,t\geq 0$.
By multiplying the function $x$ by a constant if necessary, we can assume without loss of generality that
$\rho(0)=1$($={\rm Var}(X(t))$ for all $t$).
Let $P(x)=\sum_{n=0}^N a_nx^n$ be a real-valued polynomial function, and let $d$ denotes its centered Hermite rank.

\subsection{Proof of (\ref{cv2})}
In this section, we assume that $d\geq 1$  and that $H\in(1-\frac1{2d},1)$, and our goal is to show that
$$
T^{d(1-H)-1}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
$$
converges in distribution in $C([0,1])$ to
a Hermite process of index $d$ and Hurst parameter $1-d(1-H)$, up to some multiplicative constant $C_2$.
Since $P$ has centered Hermite rank $d$, it can be rewritten as
$$
P(x)=\E[P(X(s)]+\sum_{l=d}^N b_lH_l(x),
$$
for some $b_d,\ldots,b_N\in \R$, with $b_d\neq 0$ and $H_l$ the $l$th Hermite polynomial.
As a result,
we have
\begin{eqnarray*}
\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds &=& \sum_{l=d}^N b_l (c_{H,1})^l \int_0^{Tt} I_l^B(g(s,\cdot)^{\otimes l})ds,
\end{eqnarray*}
and the desired conclusion follows thanks to Propositions \ref{assympofF} and \ref{n=1}.

\subsection{Proof of (\ref{cv1})}
In this section, we assume that $d\geq 2$ and that $H\in(\frac12,1-\frac1{2d})$, and our goal is to show that
$$
T^{-\frac12}\left\{\int_0^{Tt} \big(P(X(s))-\mathbb{E}[P(X(s))]\big)ds\right\}_{t\in [0,1]}
$$
converges in distribution in $C([0,1])$ to
a standard Brownian motion $W$, up to some multiplicative constant $C_1$.
To do so, we will rely on the Breuer-Major theorem, which asserts that the desired conclusion holds as soon as
\begin{equation}\label{tocheck}
\int_\R |\rho(s)|^d ds<\infty
\end{equation}
(see, e.g., \cite{CampeseNourdinNualart} for a continuous version of the Breuer-Major theorem).

The rest of this section   is devoted to checking  that (\ref{tocheck}) holds true. Let us first compute $\rho$:
\begin{eqnarray*}
&&\rho(t-s)=\E[X(t)X(s)]\\
&=&H(2H-1)
\iint_{\R^2}x(t-v){\bf 1}_{(-\infty,t]}(v)x(s-u){\bf 1}_{(-\infty,s]}(u)|v-u|^{2H-2}dudv\\
&=&H(2H-1)
\iint_{\R^2}x(u)x(v)|t-s-v+u|^{2H-2}dudv,
\end{eqnarray*}
with the convention that $x(u)=0$ if $u<0$.  This allows us to write
\[
\rho(s) =c_H [ \tilde{x} * (I^{2H-1} x)](s),
\]
where $\tilde {x} (u)= x(-u)$, $I^{2H-1}$  is the fractional integral operator of order $2H-1$ and $c_H$ is a constant depending on $H$.
As a consequence, applying Young's inequality and Hardy-Littlewood's inequality (see \cite[Theorem 1]{Stein}) yields
\[
\| \rho \|_{L^d(\R)} \le c_{H} \| x\|_{L^p(\R)} \|I^{2H-1} x \| _{L^q(\R)} \le c_{H,p}  \| x\|^2_{L^p(\R)},
\]
where $\frac 1d = \frac 1p +\frac 1q -1$ and $\frac 1q = \frac 1p - (2H-1)$.  This implies  $p= (H+\frac 1{2d} )^{-1}$ and we have  $\| x\|_{L^p(\R)} <\infty$,
because $p\in (1,\frac 1H)$ and   $x\in L^1(\R)\cap L^{\frac{1}{H}}(\R)$.
 The proof of (\ref{cv1}) is complete.
 \qed

 \section{The Stationary Hermite-Ornstein-Uhlenbeck process}\label{sec5bis}
We dedicate this section to the study of the extension of the Ornstein Uhlenbeck process to the case where the driving process is a Hermite process. To our knowledge, there is not much literature about this object. Among the few existing references, we mention \cite{slaoui2018limit} and \cite{nourdin2019statistical}. The special case in which the driving process is a fractional Brownian motion has been, in contrast, well studied, see for instance \cite{cheredito2003}.
In what follows, we will prove a first-order ergodic theorem for the stationary Hermite-Ornstein-Uhlenbeck process. Then, we will use Theorem $\ref{main}$ to study its second order fluctuations.

Let $\alpha>0$.  Consider the function $x(s) =e^{-\alpha s} \mathbb{I}_{s>0}$  and let $Z^{H,q}$ be a Hermite process of order $q\ge 1$ and Hurst index $H>\frac12$. Then $x\in S_L$ for all $L>0$, and we can define the stationary Hermite-Ornstein-Uhlenbeck process as:
\begin{equation}
\label{HOU}(U_t)_{t\geq 0}=\int_{-\infty}^tx(t-s)dZ^{H,q}_s.
\end{equation}

As its name suggests, this process is strongly stationary.
 We then state the following general ergodic type result.
\begin{prp}\label{ergo}
Let $(u_t)_{t\geq 0}$ be a real valued process of the form  $u_t= I^B_q(f_t)$,  where  $f_t \in L^2_s(\R^q)$  for each $t\ge 0$.
 Assume that $u$ is strongly stationary, has integrable sample paths and  satisfies, for each $1\le r\le q$,
 \[
  \|f_0\otimes_r f_{s}\|_{ L^2(\R^{2q-2r}) }\underset{s\rightarrow\infty}\longrightarrow 0.
  \]
  Then, for all measurable function such that $\mathbb{E}[|f(u_0)|]<+\infty$,
\[
\frac{1}{T}\int_0^Tf(u_s)ds\overset{a.s}{\underset{T\rightarrow\infty}\longrightarrow}\mathbb{E}\left[f(u_0)\right].
\]
\end{prp}
\begin{proof}
According to Theorem 1.3 in \cite{nourdin2016strong}, the process $u$ is strongly mixing if
for all $t>0$ and    $1\leq r\leq q$, the following convergence holds
\[
 \|f_t\otimes_r f_{t+s}\|_{L^2(\R^{2q-2r})}\underset{s\rightarrow\infty}\longrightarrow 0.
 \]
 Taking into account that $u$ is  strongly stationary, we can write
 \[
 \|f_t\otimes_r f_{t+s}\|_{L^2(\R^{2q-2r})}=\|f_0\otimes_r f_{s}\|_{L^2(\R^{2q-2r})},
 \] and the conclusion follows immediately from Birkhoff's continuous ergodic theorem.
\end{proof}

We can now particularize to the Hermite-Ornstein-Uhlenbeck process.
\begin{thm}\label{firstOrder}
Let $U$ be the Hermite-Ornstein-Uhlenbeck process defined by \eqref{HOU}. Let $f$ be a measurable function such that
$|f(x)|\leq \exp(|x|^\gamma)$ for some $\gamma<\frac{2}{q}$. Then,
\begin{equation}\lim_{T\rightarrow\infty}\frac{1}{T}\int_0^Tf(U_s)ds=\mathbb{E}\left[f(U_0)\right] \textit{ a.s}.\notag\end{equation}
\end{thm}

\begin{proof}
We shall prove that the process $U$ verifies the conditions of Proposition \ref{ergo}. We have $U_t=I_q^B(f_t)$ with
\[
f_t(x_1,\dots,x_q)=c_{H,q}\mathbb{I}_{[-\infty,t]^q}(x_1,\dots,x_q)\int_{x_1\vee \cdots \vee x_q}^te^{-\alpha(t-u)}\prod_{i=1}^q(u-x_i)^{H_0-\frac{3}{2}}du.
\]

\medskip
\noindent
\underline{\it Step 1.}  Let us first show the mixing condition, that is
\[
\lim_{s\rightarrow \infty} \|f_0\otimes_r f_{s}\|_{L^2(\R^{2q-2r})}=0
\]
 for all  $r\in\{1,\dots,q\}$.
We can write
\begin{align*}
 & f_0\otimes_rf_s(y_1,\dots, y_{2q-2r}) \\
 &=c^2_{H,q}\int_{(-\infty,0]^r}\left(\int_{x_1\vee \cdots \vee x_r\vee y_{1}\cdots\vee y_{q-r}}^0e^{\alpha u}\prod_{i=1}^r\prod_{j=1}^{q-r}(u-x_i)^{H_0-\frac{3}{2}}(u-y_j)^{H_0-\frac{3}{2}}du\right)\\
 &\quad\times\Bigg(\int_{x_1\vee \cdots \vee x_r\vee y_{q-r+1}\cdots\vee y_{2q-2r}}^{s}e^{-\alpha(s-u)}\\
 &\quad \times \prod_{i=1}^r\prod_{j=q-r+1}^{2q-2r}(u-x_i)^{H_0-\frac{3}{2}}(u-y_j)^{H_0-\frac{3}{2}}du\Bigg)dx_1  \cdots dx_r \\
 &=c^2_{H,q}\int_{y_{1}\vee\cdots \vee y_{q-r}}^0e^{\alpha u}\int_{y_{q-r+1}\vee\cdots\vee y_{2q-2r}}^{s}e^{-\alpha(s-v)}\\
 & \quad \times \left(\int_{(-\infty,u\wedge v]}(u-x)^{H_0-\frac{3}{2}}(v-x)^{H_0-\frac{3}{2}}dx\right)^r\\
 &\quad \times\prod_{j=1}^{q-r}\prod_{l=q-r+1}^{2q-2r}(u-y_j)^{H_0-\frac{3}{2}}(v-y_l)^{H_0-\frac{3}{2}}dvdu  \\
& =c^2_{H,q}\beta(H_0-\frac{1}{2}, 2-2H_0)^r\int_{y_{1}\vee \cdots \vee y_{q-r}}^0e^{\alpha u}\int_{y_{q-r+1}\vee \cdots \vee y_{2q-2r}}^{s}e^{-\alpha(s-v)}|u-v|^{r(2H_0-2)}\\
& \quad \times\prod_{j=1}^{q-r}\prod_{l=q-r+1}^{2q-2r}(u-y_j)^{H_0-\frac{3}{2}}(v-y_l)^{H_0-\frac{3}{2}}dvdu,
\end{align*}
where we used again the identity \eqref{bienutile}.
  We then have
\begin{align*}
& \|f_0\otimes_rf_{s}\|^2_{L^2(\mathbb{R}^{2q-2r})}= c^4_{H,q}\beta(H_0-\frac{1}{2}, 2-2H_0)^{2q}\\
& \quad \times \int_{(-\infty,0]^2} \int_{ (-\infty,s]^2}e^{\alpha(u+u_1)}e^{-\alpha(2s-(v+v_1))}|u-u_1|^{(q-r)(2H_0-2)}\\
& \quad \times|v-v_1|^{(q-r)(2H_0-2)}|u-v|^{r(2H_0-2)}|u_1-v_1|^{r(2H_0-2)}dv_1dvdu_1du\\
&\leq  c^4_{H,q}\beta(H_0-\frac{1}{2}, 2-2H_0)^{2q}A_0A_{s}R_{s}^2,
\end{align*}
with
\[
 A_x=\left(\int_{(-\infty,x]^2}e^{-q\alpha(2x-(u+u_1))}|u-u_1|^{q(2H_0-2)}dudu_1\right)^\frac{1}{a}
 \]
 and
 \[
R_{s}=\left(\int_{-\infty}^0\int_{-\infty}^{s}e^{-q\alpha(s-(u+v))}|u-v|^{q(2H_0-2)}dvdu\right)^\frac{1}{b},
\]
where we used the H\"older inequality with $a=\frac{q}{q-r},b=\frac{q}{r}$.
 Making the change of variable $x-u=v$, $x-u_1=v_1$, we obtain
 \begin{align*}
& \int_{(-\infty,x]^2}e^{-q\alpha(2x-(u+u_1))}|u-u_1|^{q(2H_0-2)}dudu_1\\
& \quad =\int_{[0,\infty)^2}e^{-q\alpha( v+v_1)}|v-v_1|^{q(2H_0-2)}dudu_1<\infty.
\end{align*}
 On the other hand, we have $q(2H_0-2)=2H-2$, so

\begin{align*}
R^b_s={\rm Cov}(U^H_0,U^H_s)
\end{align*}
where $U^H$ is a stationary Ornstein Uhlenbeck process driven by a fractional Brownian motion of index $H$ (and with $\alpha^H=q\alpha$). According to \cite[Lemma 2.2]{cheredito2003}, one has $R^b_s=O_{s\rightarrow\infty}(s^{-H})$, implying in turn that
 $ \lim_{s\rightarrow\infty}R_{s}=0$
and concluding the proof of the mixing condition.

 \medskip
\noindent
\underline{\it Step 2.}  We now show the integrability condition $\E[ | f(U_0)|] <\infty$. From the results of   \cite{cheredito2003},
we have $\mathbb{E}[U_0^2]=\frac{1}{\alpha^{2H}}\Gamma(2H)$. A power series development yields
\[
\mathbb{E}[|f(U_0)|]\leq \sum_{k=0}^{\infty}\frac{1}{k!}\mathbb{E}[|U_0|^{\gamma k}],
\]
where $U_0$ is an element of the $q$th Wiener chaos. By the hypercontractivity property, for all $ k\geq \frac{2}{\gamma}$,
\[
\mathbb{E}[|U_0|^{\gamma k}]\leq g(k):= (k-1)^{\frac{\gamma qk}{2}}\left(\frac{1}{\alpha^{2H}}\Gamma(2H)\right)^\frac{\gamma k}{2}.
\]
 Stirling formula allows us to write
  \begin{equation}\frac{g(k)}{k!}\sim_{k\rightarrow\infty}\frac{(k-1)^{\frac{\gamma qk}{2}}}{k^{k}}\frac{\left(\frac{1}{\alpha^{2H}}\Gamma(2H)\right)^\frac{\gamma k}{2}e^k}{\sqrt{2\pi k}},\end{equation}
and the associated series converges if $\gamma q<2$.
\end{proof}

 The next result analyzes the fluctuations in the   ergodic theorem proved in Theorem  \ref{firstOrder}.

\begin{thm}\label{secondOrder}
\textit{(A)} {\rm [Case $q=1$]} Let $f$ be in $L^2(\R, \gamma)$ for $\gamma=\mathcal N(0,\frac{\Gamma(2H)}{\alpha^{2H}})$.
We denote by  $(a_i)_{i\geq 0}$ the coefficients of $f$ in its Hermite expansion, and we let $d$ be the centered Hermite rank of $f$. Then,
\begin{itemize}
  \item if $\frac{1}{2}<H<1-\frac{1}{2d}$,
\begin{equation*}
\frac{1}{\sqrt{T}}\int_0^{Tt}\left(f(U_s)-\mathbb{E}[f(U_0)]\right)ds\overset{f.d.d}{\underset{T\rightarrow\infty}\longrightarrow} c_{f,H}W_t,
\end{equation*}
   \item if $H=1-\frac{1}{2d}$,
\begin{equation*}
\frac{1}{\sqrt{T\log{T}}}\int_0^{Tt}\left(f(U_s)-\mathbb{E}[f(U_0)]\right)ds\overset{f.d.d}{\underset{T\rightarrow\infty}\longrightarrow}c_{f,H}W_t,
\end{equation*}
   \item if $H>1-\frac{1}{2d}$,
\begin{equation*}
T^{q(1-H)-1}\int_0^{Tt}\left(f(U_s)-\mathbb{E}[f(U_0)]\right)ds\overset{f.d.d}{\underset{T\rightarrow\infty}\longrightarrow} c_{f,H}Z^{d,H}_t,
\end{equation*}
\end{itemize}
where $Z^{d,H}$ is a Hermite process of order $d$ and index $d(H-1)+1$, $W$ is a Brownian motion and
\begin{equation}
c_{f,H}
=
\left\{
\begin{array}{cl}
\sqrt{\sum_{k\geq d}k!a^2_k\int_{\mathbb{R}_+}|\rho(s)|^k} & \textit{ if } \,\, H<1-\frac{1}{2d}\\
a_d\sqrt{d!\frac{3}{16\alpha^2}}      & \textit{ if } \,\, H=1-\frac{1}{2d}\\
a_d\sqrt{d!\frac{H^d\Gamma(2H)^d}{\alpha^{2Hd}}}      & \textit{ if } \,\, H>1-\frac{1}{2d}
\end{array}
\right.
\end{equation}
with
\[
\rho(s)=\mathbb{E}[U_sU_0]=\int_{-\infty}^0\int_{-\infty}^se^{-\alpha(s-(u+v))}|u-v|^{2H-2}dudv.
\]
Moreover, if $f\in L^p(\R,\gamma)$ for some $p>2$, the previous convergences holds true in the Banach space $C([0,1])$.\\
\textit{(B)} {\rm [Case $q>1$]} Let $P$ be a real valued polynomial. Then, the conclusions of Theorem \ref{main} apply to $U$.
\end{thm}
\begin{proof}
Except for $H=1-\frac{1}{2d}$ in Part A, this is a direct consequence of Theorems \ref{FBM} and \ref{main}.
The convergence in the critical case
can be checked through easy but tedious computations, by reducing to the case where $f$ is the $d$th Hermite polynomial. Details are left to the reader.
\end{proof}

\section{Appendix}\label{sec6}
In this section we present two technical lemmas that  play an important  role along the paper.
First, we shall reproduce a very useful result from \cite{Taqqu2}:

\begin{thm}[Power Counting Theorem]\label{powercounting}
Let $T=\{M_{1},\ldots,M_K\}$ a set of linear functionals on $\mathbb{R}^n$, $\{f_1,\ldots, f_K\}$ a set of real measurable functions
on $\R^n$ such that  there exist  real numbers $(a_i,b_i,\mu_i,\nu_i)_{1\leq i\leq K}$,  satisfying for each $i=1, \dots, K$,
\begin{eqnarray*}
&& 0<a_i\leq b_i,\\
&& |f_i(x)| \le |x|^{\mu_i} \textit{ if } |x|\leq a_i,\\
&& |f_i(x)| \le  |x|^{\nu_i} \textit{ if } |x|\geq b_i,\\
&&  f_i \textit{ is bounded over } [a_i,b_i].
\end{eqnarray*}
For   a linearly independent subset of $W$ of  $T$, we write $S_T(W)={\rm Span}(M)\cap T$. We also define
\begin{eqnarray*}
d_0(W)={\rm dim}({\rm Span}(W))+\sum_{i: M_i \in S_T(W)}\mu_i, \\
d_\infty(W)=n-{\rm dim}({\rm Span}(W))+\sum_{i:M_i \in T\setminus S_T(W)}\nu_i.
\end{eqnarray*}
Assume ${\rm dim}({\rm Span}(T))=n$. Then, the two conditions
$(a):d_0(W)>0$ for all linearly independent subsets $W\subset T$, $(b):d_{\infty}(W')<0$ for all linearly independent proper subsets $W'\subset T$,  imply
\begin{equation}  \label{finite}
\int_{\mathbb{R}^n}\prod_{i=1}^K|f_i(M_i(x))|dx<\infty
\end{equation}
Moreover, assume that $|f_i(x)| = |x|^{\mu_i} \textit{ if } |x|\leq a_i$, Then\\
 $ \int_{[-1,1]^n}\prod_{i=1}^K|f_i(M_i(x))|dx<\infty$,  if an only if
   for any linearly independent subset $W\subset T$ condition $(a)$ holds.
\end{thm}

The next lemma is an application of the Hardy-Littlewood-Sobolev inequality,
\begin{lem}\label{HLS-lemme}
Fix $n,q\geq 2$ and $\alpha\in A_{n,q}$. Recall    $H$ and $H_0$ from (\ref{hzero}).
Assume $x\in L^1(\mathbb{R})\cap L^{\frac1H}(\mathbb{R})$
Then
$$\int_{\mathbb{R}^n} \prod_{k=1}^n|x(\eta_k)| \prod_{1\leq i<j\leq n}|\eta_i-\eta_j|^{(2H_0-2)\alpha_{ij}}  d\eta_1\ldots d\eta_n<\infty.$$
\end{lem}
\begin{proof}
We are going to use
 the multilinear Hardy-Littlewood-Sobolev inequality, that we recall here for the convenience of the reader (see \cite[Theorem 6]{Beckner}):
if $f:\mathbb{R}\to\mathbb{R}$ is a measurable function, if $p\in(1,n)$ and if the $\gamma_{ij}\in(0,1)$ are such that $\sum_{1\leq i<j\leq n}\gamma_{ij}=1-\frac 1p$, then there exists $c_{p,\gamma}>0$ such that
\begin{equation}\label{HLS}
\int_{\mathbb{R}^n} \prod_{k=1}^n|f(u_k)|  \prod_{1\leq i<j\leq n}|u_i-u_j|^{-\gamma_{ij}}du_1\ldots du_n\leq c_{p,\gamma} \left(\int_{\mathbb{R}}|f(u)|^pdu\right)^{\frac{n}p}.
\end{equation}

Set $p=1/(1-(1-H)\frac{2|\alpha|}{nq})$.
Since $2|\alpha|\leq nq$, we have that $p>1$.
On the other hand, since $H>\frac12$, one has $nH>\frac{n}2\geq 1$; this implies that
$(1-H)\frac{2|\alpha|}{q}<(1-H)n<n-1$, that is, $p<n$.
Moreover, set $\gamma_{ij}=(2-2H_0)\alpha_{ij}=(1-H)\frac{2\alpha_{ij}}{q}\in(0,1)$;
we have
$
\sum_{1\leq i<j\leq n}\gamma_{ij}=2(1-H)\frac{|\alpha|}{q}\leq (1-H)n<n-1.
$
We deduce from (\ref{HLS}) that
$$
\int_{\mathbb{R}^n} \prod_{k=1}^n|x(\eta_k)| \prod_{1\leq i<j\leq n}|\eta_i-\eta_j|^{(2H_0-2)\alpha_{ij}}  d\eta_1\ldots d\eta_n\leq c_{p,\gamma}
\left(\int_{-\infty}^\infty|x(u)|^pdu\right)^{\frac{n}{p}}.
$$
But $p\in(1,\frac1H)$ and $x\in L^1(\mathbb{R})\cap L^{\frac1H}(\mathbb{R})$, so the claim follows.
\end{proof}

\medskip
\noindent
{\bf Acknowledgments}: We are grateful to Pawel Wolff for making us aware
of the multivariate Hardy-Littlewood-Sobolev inequality from \cite{Beckner}.


TITLE: Is the differentiation operator normal?
QUESTION [2 upvotes]: In the space of polynomials (degree not higher than n) , the scalar product is given by the formula:
$$(f,g)=\int_{0}^{1}f(x)g(x)dx$$
is the differentiation operator normal?

REPLY [4 votes]: This is never true for $n\geq 1$:
Let $D$ be the differential operator and $D^*$ it's adjoint.
Then
$$\langle x,D^*(1)\rangle =\langle D(x),1\rangle=\langle 1,1\rangle=1$$
so $D^*(1)\neq 0$ and hence
$$\langle DD^*(1),1\rangle=\langle D^*(1),D^*(1)\rangle=\|D^*(1)\|^2\neq 0$$
which then implies $DD^*(1)\neq 0$.
But $D^*D(1)=D^*(0)=0$, so $D^*D\neq DD^*$.


TITLE: Need help understanding this text
QUESTION [1 upvotes]: I was reading about Relations and Partial Orders and it states:

Corollary 7.9.4. Every partially ordered set with $n$ elements has a
  chain of size greater than $\sqrt n$ or an antichain of size at least $\sqrt n$.

I know the reasoning behind this, however I was not able to understand the text following this:

As an application, consider a permutation of the numbers from $1$ to $n$
  arranged as a sequence from left to right on a line. Corollary 7.9.4
  can be used to show that there must be a $\sqrt n$ length subsequence of
  these numbers that is completely increasing or completely decreasing as you move from left to right. For example, the sequence $7, 8, 9, 4, 5, 6, 1, 2, 3$
  has an increasing subsequence of length 3 (for example, 7, 8, 9) and a decreasing subsequence of length 3 (for example, 9, 6, 3). The proof of this result is left as an exercise that will test your ability to find the right partial order on the numbers in
  the sequence.

I don't understand how Corollary 7.9.4. can be used to show that there must be a $\sqrt n$ length subsequence of these numbers that is completely increasing or completely decreasing as you move from left to right.
And how is this permutation of numbers from $1$ to $n$ relatable to POSETs? POSET consists of a set and a partial order(a relationship of a special kind). So do we consider the numbers in the sequence as the elements of the set? And when they say that "proof of this result is left as an exercise that will test your ability to find the right partial order on the numbers in the sequence", do they mean the example sequence which they mentioned above or any permutation sequence in general?
And one last thing, the example sequence they gave: $7, 8, 9, 4, 5, 6, 1, 2, 3$, will this be considered as topological sort for a particular POSET?
PS:

An antichain in a poset is a set of elements such that any two
  elements in the set are incomparable.

REPLY [0 votes]: Given a permutation $\sigma$ on the set $S_n=\{1,2,\ldots,n\}$ define the following order relation on $S_n$:
$$x \lesssim y \quad\text{ iff }\quad x \leq y \quad\text{and}\quad \sigma^{-1}(x) \leq \sigma^{-1}(y).$$
So you'll have $x \lesssim y$ whenever $x \leq y$ and $x$ appears first in the sequence of images of $\sigma$, when going from left to right.
Hence, you have a chain $a_1 \lesssim \cdots \lesssim a_k$ if $a_1, \ldots, a_k$ is a increasing sub-sequence of $\sigma(1), \ldots, \sigma(n)$.
And you'll have an anti-chain $b_1, \ldots, b_m$ if $b_1 \leq \cdots \leq b_m$ but $\sigma^{-1}(b_1) \geq \cdots \geq \sigma^{-1}(b_m)$.
By the corollary you mention, the poset has at least one $\sqrt{n}$-elements chain or anti-chain, and therefore, the sequence $(\sigma(1), \ldots, \sigma(n))$ has at least one $\sqrt{n}$-elements increasing sub-sequence or a $\sqrt{n}$-elements decreasing one.

In the example you refer to, the result is a poset consisting of the union of three three-element chains:
$$1 \lesssim 2 \lesssim 3, \quad 4 \lesssim 5\lesssim 6, \quad\text{and}\quad 7 \lesssim 8 \lesssim 9.$$
It is immediate that, indeed, $1,2,3$, and $4,5,6$ and $7,8,9$ are increasing sequences (and they are sub-sequences of $(7,8,9,4,5,6,1,2,3)$).
On the other hand, if you pick one element from each sequence, from the third to the first, it forms a decreasing sub-sequence of $(7,8,9,4,5,6,1,2,3)$.
If you play around a little bit with different permutations (even with $n \leq 9$), you'll see that the poset can be much less trivial than this one.
But the real point is that one can derive the conclusion for the general case using the result about posets.


TITLE: Is $f(t)=(\cos(t),\sin(t))$ a function?
QUESTION [3 upvotes]: In the Linear Algebra book we're using (Linear Algebra with Applications, Bretscher, p.129), the author defines this as the function of the unit circle.
I understand why the equation of a circle can't be a function. In this case however, I could see how despite failing the vertical line test, $f(t)=(\cos(t),\sin(t))$ still outputs a unique value for any $t$ and is therefore technically a function.
Is that the rationale, or is $f(t)=(\cos(t),\sin(t))$ not really a function despite what the author states?

REPLY [3 votes]: This is just parametric equation of a circle $f(t)=\left[\begin{matrix}\cos t\\\sin t\end{matrix}\right]$ http://www.mathopenref.com/coordparamcircle.html, it's a function from $\mathbb{R}$ to $\mathbb{R}^2$ and its range is the unit circle centered at the origin.


TITLE: Kind of linear property on a function $f: X \to Y$
QUESTION [0 upvotes]: A function $f: X \to Y$ is called linear iff
$f(x_1+x_2) = f(x_1) + f(x_2)$
It is what I learned.
Is there any concept like 'linear' defined on the functions whose input and output are sets.
A function $f: X \to Y$ is called 'ABCD' iff
   $f(X1 \cup X2) = f(X1) \cup f(X2)$
Is there any concept corresponding to 'ABCD' here?
// Thanks, lulu
I clarify here:
f is defined on some fixed set and that X_i denotes subsets of that set.

REPLY [1 votes]: If $f(x_1+x_2)=f(x_1)+f(x_2)$ then $f$ is called additive and not linear !

If $X$ and $Y$ are vector spaces over a field $F$ and we have additional
$f( \alpha x)= \alpha f(x)$ for all $ \alpha \in F$ and all $x \in X$,
then f is called linear.

For each function $f:X \to Y$ we have

$f(X_1 \cup X_2)=f(X_1) \cup f(X_2)$ for all subsets $X_1,X_2$ of $X$


TITLE: Example of an integral operator satisfying Holmgren's condition but not compact
QUESTION [1 upvotes]: Let $F$ be an integral operator with the kernel $K(s,t)$. This means that $F(f)(t)=\int K(s,t)f(s)ds$ for square integrable functions $f(s)$.
Here $K$ satisfies the condition $[sup_{s} \int \mid K(s,t) \mid dt] \cdot [sup_{t} \int \mid K(s,t) \mid ds] < \infty$.
Is there an example of $K(s,t)$ such that the operator $F$ is not compact as a mapping from some $L^2$ space to another $L^2$ space? I cannot find one myself.

REPLY [0 votes]: The identity operator on $\ell^2$ is not compact, but is an integral operator whose kernel has the property in the question.


TITLE: Find the sum of the first $n$ positive integers
QUESTION [0 upvotes]: I just wanted to double-check my work on this problem. The problem states,
Let $a$ and $k$ be any positive integers. Find the sum $a+(a+1)+(a+2)+\cdots+(a+k)$.
Saying the summation is equal to ${n(n+1)\over 2}$, I got the answer ${(a+k)(a+k+1)\over 2}$, or ${a^2 \over 2}+ak+{a \over 2}+{k^2 \over2}+{k \over 2}$.
Does this make sense?

REPLY [0 votes]: Sorry, but no, your method does not make sense.
As undoubtedly many answers will show, you can use that formula, but not in the way you use it.
The $\frac{n(n+1)}{2}$ formula is for the sum of the $n$ numbers $1$ through $n$, so your $\frac{(a+k)(a+k+1)}{2}$ would be the sum of the numbers $1$ through $a+k$, rather than $a$ through $a+k$


TITLE: Discrete Signals
QUESTION [0 upvotes]: Can anyone please tell me
What is the period of $\cos(n) u(n)$?
Here Cosine is multiplied by a unit step sequence .
I am just learning the basics of signals now and I have found that $\cos(t) u(t)$ is aperiodic and no fundamental period in continuous domain, as Heaviside function is discontinuous at zero .what is the period in case of discrete where there is no jump in the sequence.

REPLY [1 votes]: Let $\;f(n) := \cos(n)u(n)\;$ where $u(x)$ is the unit step function then $\;f(n) = 0\;$ for all $\;n<0.$ If $f(n)$ were periodic  then it would be the zero sequence which it isn't. Thus, $f(n)$ is not periodic. In other words, the aperiodicity of $f$ comes from its zero values and not in its discontinuity.


TITLE: Double integral bounded by square
QUESTION [0 upvotes]: I have this double integral and I am not sure how to use the bound region to generate my limit  bounds..
the double integral is y-2x^2 dA      and it is bounded by |x| + |y| = 1
What do I do to start integrating this and how would I get the bound values from this? I am stuck on every part of this from integrating to coming up with the bounds.

REPLY [1 votes]: It will help to draw a picture of your domain. In this case you can choose to integrate over $y$ first, and then (as you will be able to see from a plot), $y$ ranges from 0 to $|x|-1$. You can then split up your integral into 2 domains in x, $(-1,0)$, and $(0,1)$, You then get:
$\int_{-1}^0 \int_0^{x+1} f(x,y) dy dx$ on $(-1,0)$,
$\int_0^1 \int_0^{1-x} f(x,y) dy dx$ on $(0,1)$,
where $f(x,y) = y-2x^2$ and $dy dx$ means you integrate over y first, then x.The idea here is that you integrate up to one variable to get your function in terms of one variable then you have an absolute integral.


TITLE: Gradient of an Electric Dipole
QUESTION [0 upvotes]: This may seem like a physics question but I am trying to solve it from a mathematics book actually. An electric dipole is given as:
$$\psi(\overrightarrow r) = \frac{\overrightarrow p ⋅ \overrightarrow r}{4\pi\epsilon_0r^3} $$
Now I need to prove that
$$\overrightarrow ∇\psi(\overrightarrow r) =  \frac{\overrightarrow p - 3\hat{r}(\overrightarrow p ⋅ \hat r)}{4\pi\epsilon_0r^3}$$
I used $∇$ in spherical coordinates and put $\overrightarrow r = r × \hat{r}$. No matter what I do I can only find
$$ \overrightarrow ∇\psi(\overrightarrow r) = \frac{-3\overrightarrow p}{4\pi\epsilon_0r^3}$$
Could you please help me with this? Thank you very much in advance!!

REPLY [1 votes]: To reduce some of the visual clutter in the problem statement, define 2 new variables
$$\eqalign{
 a &= \frac{p}{4\pi\epsilon_0} \cr
 \lambda &= \|r\| \cr
}$$
Note that
$$\eqalign{
 \lambda^2 &= r\cdot r \cr
 \lambda\,d\lambda &= r\cdot dr \cr\cr
}$$
Write the function in terms of these new variables, then find its differential and gradient
$$\eqalign{
 \psi &= \frac{a\cdot r}{\lambda^3} \cr
\cr
d\psi
 &= \frac{a\cdot dr}{\lambda^3} -\frac{3(a\cdot r)\,d\lambda}{\lambda^4} \cr
 &= \frac{a\cdot dr}{\lambda^3} -\frac{3(a\cdot r)\,(r\cdot dr)}{\lambda^5} \cr
 &= \frac{a-(3a\cdot{\hat r})\,{\hat r}}{\lambda^3}\cdot dr \cr
\cr
\nabla\psi
  &= \frac{a-(3a\cdot {\hat r})\,{\hat r}}{\lambda^3} \cr
  &= (I-3{\hat r}{\hat r})\cdot a\lambda^{-3} \cr\cr
}$$


TITLE: prove $A \times B = B \times A \iff (A=B)\lor(A=\emptyset)\lor(B=\emptyset)$
QUESTION [0 upvotes]: had to prove the above mentioned iff, but I think I have something missing. would appreciate your advice.

REPLY [3 votes]: What you wrote is okay. The first part is trivial and the second part is correct, but I would prefer:
Assume NOT the right side. That is $A\neq B$ and neither $A$ nor $B$ are empty. Then without loss of generality, there is $a\in A$ such that $a\not\in B$ and there is some $b\in B$.
Then the element $(a,b) \in A\times B$ but $(a,b)\not\in B\times A$.


TITLE: Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$
QUESTION [6 upvotes]: The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has
(A) exactly two distinct real roots
(B) exactly three distinct real roots
(C) exactly four distinct real roots
(D) six distinct real roots

REPLY [8 votes]: You have:
$f(x)=x^6-5x^4+16x^2-72x+9$
$f'(x)=6x^5-20x^3+32x-72$
$f''(x)=30x^4-60x^2+32$
If you notice that
$$f''(x)=30(x^4-2x^2+1)+2=30(x^2-1)^2+2 \ge 2 >0,$$
you can see that $f'(x)$ is strictly increasing. Together with $\lim\limits_{x\to-\infty} f'(x)=-\infty$ and $\lim\limits_{x\to\infty} f'(x)=\infty$ this implies that there is exactly one root $x_0$ of $f'(x)$.
Thus $f(x)$ is decreasing on $(-\infty,x_0)$ and increasing on $(x_0,\infty)$.
Since $\lim\limits_{x\to\infty} f(x)=\infty$ and $f(1)=1-5+16-72+9=-51$, we see that $f(x)$ has both positive and negative values.
Thus $f(x)$ must have two real roots, one of them in the interval $(-\infty,x_0)$ and another one in the interval $(x_0,\infty)$.

You can check the behavior of $f(x)$ and the behavior of $f'(x)$ at WolframAlpha.


TITLE: Lie powers of a graded vector space and Klyachko's theorem
QUESTION [6 upvotes]: Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by the number of generators (the generators $V$ sit in degree 1, and the Lie bracket itself has degree zero in this sense). So it has a decomposition by degrees, $L(V) = \sum_{n=1}^\infty L^n(V)$. The homogeneous components $L^n(V)$ may be called (super) Lie powers of $V$ (by analogy with symmetric or alternating powers) and have an embedding $L^n(V) \subset V^{\otimes n}$, where the Lie (super)commutators are sent to tensor (super)commutators, $[a,b] = a\otimes b - (-)^{|a||b|} b \otimes a$.
Q: What is the $\mathrm{GL}(V)$-representation theoretic description of $L^n(V)$?
When $V$ is purely even (has no odd component), the question is answered by Klyachko's theorem (as explained in the answers to MO187545). So basically, I'm asking: What is the super Lie algebra analog of Klyachko's theorem? And what is a reference that clearly states it?

REPLY [1 votes]: I have found a more narrow reference. It deals quite specifically with my question (confirming Qiauchu Yuan's answer), as well as some generalizations thereof:

Kwon, Jae-Hoon, Free Lie-superalgebras and the representations of $\mathfrak{gl}(m,n)$ and $\mathfrak{q}(n)$ (doi), J. Korean Math. Soc. 42, No. 2, 365-386 (2005). ZBL1168.17301.