Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0}{Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1)H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel gg, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels gg can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}{X(n)}. In addition, we consider a fractionally-filtered version Zβ(t)Zβ(t) of Z(t)Z(t), which allows H∈(0,1/2)H∈(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.