Limit theorems for integral functionals of Hermite-driven processes

Abstract

Consider a moving average process $X$ of the form $X(t)={\int }_{-\infty }^{t}x(t-u)d{Z}_u$, $t\ge 0$, where $Z$ is a (non Gaussian) Hermite process of order $q\ge 2$ 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 Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.

25 pages