Limit theorems for integral functionals of Hermite-driven processes

Abstract

Consider a moving average process X of the form X(t) = t −∞ ϕ(t − u)dZu, t ≥ 0, where Z is a (non Gaussian) Hermite process of order q ≥ 2 and ϕ : R+ → R is sufficiently integrable. This paper investigates the fluctuations, as T → ∞, of integral functionals of the form t → T t 0 P (X(s))ds, in the case where P is any given polynomial function. It extends a study initiated in [21], where only the quadratic case P (x) = x 2 and the convergence in the sense of finite-dimensional distributions were considered.