Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes \{Z(t),t\ge 0\} called generalized Hermite process. They have stationary increments, are defined on a Wiener chaos with Hurst index H\in (1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g, called "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\in (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.This work was partially supported by the NSF grants DMS-1007616 and DMS-1309009 at Boston University. (DMS-1007616 - NSF at Boston University; DMS-1309009 - NSF at Boston University)Accepted manuscrip