Limit theorems for integral functionals of Hermite-driven processes

Abstract

Consider a moving average process X of the form $\mathit{X}(\mathit{t})={\int }_{-\infty }^{\mathit{t}}\mathit{\phi }(\mathit{t}-\mathit{u})\mathit{d}{\mathit{Z}}_{\mathit{u}}$, $\mathit{t}\ge 0$, where Z is a (non Gaussian) Hermite process of order $\mathit{q}\ge 2$ and $\mathit{\phi }:{\mathbb{R}}_{+}\to \mathbb{R}$ is sufficiently integrable. This paper investigates the fluctuations, as $\mathit{T}\to \infty$, of integral functionals of the form $\mathit{t}\mapsto {\int }_0^{\mathit{T}\mathit{t}}\mathit{P}(\mathit{X}(\mathit{s}))\mathit{d}\mathit{s}$, in the case where P is any given polynomial function. It extends a study initiated in (Stoch. Dyn. 18 (2018) 1850028, 18), where only the quadratic case $\mathit{P}(\mathit{x})={\mathit{x}}^2$ and the convergence in the sense of finite-dimensional distributions were considered.