Limit theorems for integral functionals of Hermite-driven processes

Abstract

Consider a moving average process X of the form X(t)=∫−∞tφ(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↦∫0TtP(X(s))ds, 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 P(x)=x2 and the convergence in the sense of finite-dimensional distributions were considered.