Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes

TL;DR

This paper develops an asymptotic expansion and a central limit theorem for quadratic variations of Gaussian processes, enhancing understanding of their convergence properties and applications to fractional Brownian motions.

Contribution

It introduces new asymptotic expansion and CLT results for quadratic variations, extending previous convergence results for Gaussian processes.

Findings

Deterministic asymptotic expansion for quadratic variations

Central limit theorem for quadratic variations

Application to identify fractional Brownian motions

Abstract

Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion.