Limit theorems for additive functionals of some self-similar Gaussian processes

Abstract

Under certain mild conditions, limit theorems for additive functionals of some $d$-dimensional self-similar Gaussian processes are obtained. These limit theorems work for general Gaussian processes including fractional Brownian motions, sub-fractional Brownian motions and bi-fractional Brownian motions. To prove these results, we use the method of moments and an enhanced chaining argument. The Gaussian processes under consideration are required to satisfy certain strong local nondeterminism property. A tractable sufficient condition for the strong local nondeterminism property is given and it only relays on the covariance functions of the Gaussian processes. Moreover, we give a sufficient condition for the distribution function of a random vector to be determined by its moments.