A central limit theorem for a weighted power variation of a Gaussian process*

Abstract

Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t − s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V n corresponding to a regular partition π n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V n as the mesh of π n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes.