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 ∈ ( 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 β ( t ) of Z ( t ) , which allows 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.