Non-Central Limit Theorem for Quadratic Functionals of Hermite-Driven Long Memory Moving Average Processes

Abstract

Let $(Z_t^{(q,H)}{)}_{t\ge 0}$ denote a Hermite process of order $q\ge 1$ and self-similarity parameter $H\in (\frac{1}{2},1)$. Consider the Hermite-driven moving average process $$X_t^{(q,H)}={\int }_0^{t}x(t-u)d{Z}^{(q,H)}(u),t\ge 0.$$ In the special case of $x(u)=e^{-\theta u},\theta >0$, $X$ is the non-stationary Hermite Ornstein-Uhlenbeck process of order $q$. Under suitable integrability conditions on the kernel $x$, we prove that as $T\to \infty$, the normalized quadratic functional $$G_T^{(q,H)}(t)=\frac{1}{T^{2H_0-1}}{\int }_0^{Tt}\left(\right(X_s^{(q,H)}{)}^2-E\left[\right(X_s^{(q,H)}{)}^2\left]\right)ds,t\ge 0,$$ where $H_0=1+(H-1)/q$, converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter $H^{\prime }=1+(2H-2)/q$, up to a multiplicative constant, irrespective of self-similarity parameter whenever $q\ge 2$. In the Gaussian case $(q=1)$, our result complements the study started by Nourdin \textit{et al} in arXiv:1502.03369, where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener-Itô integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.

Accepted for publication in Stoch. Dyn

References in corpus (2)

• Analysis of the Rosenblatt process
• Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders