Non-central limit theorems for quadratic functionals of Hermite-driven long memory moving average processes

Abstract

Let [Formula: see text] denote a Hermite process of order [Formula: see text] and self-similarity parameter [Formula: see text]. Consider the Hermite–driven moving average process [Formula: see text] In the special case of [Formula: see text], [Formula: see text] is the non-stationary Hermite Ornstein–Uhlenbeck process of order [Formula: see text]. Under suitable integrability conditions on the kernel [Formula: see text], we prove that as [Formula: see text], the normalized quadratic functional [Formula: see text] where [Formula: see text], converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter [Formula: see text], up to a multiplicative constant, irrespective of self-similarity parameter whenever [Formula: see text]. In the Gaussian case [Formula: see text], our result complements the study started by Nourdin et al. in [10], where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener–Itô integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.