Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t), t >= 0) called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H is an element of (1/2, 1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g, called the "generalized Hermite kernel", which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X (n)). In addition, we consider a fractionally-filtered version Z(beta) (t) of Z(t), which allows H is an element of (0, 1/2). Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems. (C) 2014 Elsevier B.V. All rights reserved.