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

Let rho be a real-valued function on [0, T], and let LSI(rho) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance sigma(s, t) is close to the values rho(|t - s|) as t -> s uniformly in s a (0, T]. For a Gaussian processesGfrom LSI(rho), we consider a power variation V (n) corresponding to a regular partition pi (n) of [0, T] and weighted by values of rho(center dot). Under suitable hypotheses on G, we prove that a central limit theorem holds for V (n) as the mesh of pi (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.