# Long-memory process and convergence of finite dimensional distributions

We assume that $(X_t)_{t\in\mathbb{N}}$ is a stationary sequence of standard normal random variables such that for the autocovariance function holds $\gamma _X (k)\sim Ck^{2d-1}$ with $d \in (0,1/2)$ and $C>0$. Let $Y_t:=G(X_t)$ where $G$ is a real function such that $\mathbb{E}(G(X_1))=0$ and $\mathbb{E}(G(X_1)^2)<\infty$.

We call $m:=\min_{k\in\mathbb{N}_{\geq1}}\operatorname{E}\left[G(Z)H_k(Z)\right]$ the hermite rank of $G$, where $(H_k(x))_{k\in\mathbb{N}}$ denotes the sequence of hermite polynomials. If the hermite rank $m$ of $G$ is $1$ we have that \begin{align*} \underbrace{n^{-(\frac{1}{2}+d))}L^{-\frac{1}{2}}}_{=:v_l^{-1}}\cdot\sum_{t=1}^{[nu]}G(X_t)\Rightarrow J(1)B_{H}(u)\text{ }(u\in\left[0,1\right]). \end{align*} Where $L:=C\cdot C_1$ with $C_1:=\frac{1}{d(2d+1)}$. By $B_{H}$ we note the Brownian Bridge and by $\Rightarrow$ convergence in $D\left[0,1\right]$. $H$ is given by $H:=d+\frac{1}{2}$.

We know from above that for each $i\in\mathbb{N}$ holds that $T_{i,l}:=v_l^{-1}\cdot\sum_{t=i}^{i+l-1}G(X_t)$ converge in distribution to a standard normal random variable $Z_i$ for $l\rightarrow\infty$. Now we can choose by Kolmogorovs theorem a stationary Gaussian prozess $(Z_i)_{i\in\mathbb{N}}$ with $\operatorname{E}[Z_i]=0$ and $\operatorname{Var}(Z_i)=1$. Further it should hold that if we consider $l\in\mathbb{N}$ as a function of $k$ i.e. $l:\mathbb{N}\rightarrow\mathbb{N}$ such that $\frac{l(k)}{k}\xrightarrow{k\rightarrow\infty}0$ that $\operatorname{Cov}(T_{i,l(k)},T_{k,l(k)})\xrightarrow{k\rightarrow\infty}\operatorname{Cov}(Z_i,Z_k)$ holds.

What I want to say is that the asymptotic behaviour of the covariance of $(T_{i,l})_{i\in\mathbb{N}}$ for $l\rightarrow\infty$ is the same as the asymptotic behaviour of the covariance function of $(Z_i)_{i\in\mathbb{N}}$ i.e. for $k\rightarrow\infty$ holds that $\gamma_{T_l}(k):=\operatorname{Cov}(T_{i,l(k)},T_{i+k,l(k)})\sim \operatorname{Cov}(Z_i,Z_{i+k})=:\gamma_{Z}(k)$ .

And now my question: Is it somehow possible to follow that the finite dimensional distributions of $(T_{i,l})_{i\in\mathbb{N}}$ converge to the corresponding finite dimensional distributions of $(Z_i)_{i\in\mathbb{N}}$ in this particular case? I.e. $P(T_{i_1,l}\leq x_1 ,...,T_{i_n,l}\leq x_n)\rightarrow P(Z_{i_1}\leq x_1 ,...,Z_{i_n}\leq x_n)$ with $i_1,...,i_n\in\mathbb{N}$ and $x_1,...,x_n\in\mathbb{R}$.

Actually that's more that I need. It would be enough if I could show that $$\operatorname{Cov}(1\{T_{i,l(k)}\leq u\},1\{T_{i+k,l(k)}\leq u\})\sim \operatorname{Cov}(1\{Z_i\leq u\},1\{Z_{i+k}\leq u\})$$for $k\rightarrow\infty$ or if I could actually compute $$\operatorname{Cov}(1\{T_{i,l(k)}\leq u\},1\{T_{i+k,l(k)}\leq u\})$$ asymptotically, but I have no idea how to do that. Computing $\operatorname{Cov}(1\{Z_i\leq u\},1\{Z_{i+k}\leq u\})$ asymptotically is much easier because one can expand $1\{Z_i\leq u\}$ in terms of hermite polynomials. Are multivariate hermite polynomials an option to compute $\operatorname{Cov}(1\{T_{i,l(k)}\leq u\},1\{T_{i+k,l(k)}\leq u\})$? Any ideas?