Limit theorems for integral functionals of Hermite-driven processes

TL;DR

This paper studies the asymptotic behavior of integral functionals of Hermite-driven processes, extending previous work to general polynomial functions and analyzing their fluctuations as the observation window grows large.

Contribution

It generalizes prior results from quadratic functionals to arbitrary polynomial functions, providing new limit theorems for Hermite-driven processes.

Findings

Derived limit theorems for integral functionals with polynomial P

Extended previous quadratic case to general polynomials

Analyzed fluctuations as T approaches infinity

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 Tran (2018), where only the quadratic case $P(x)=x^2$ and the convergence in the sense of finite-dimensional distributions were considered.