Limit theorems for functionals of two independent Gaussian processes

TL;DR

This paper establishes limit theorems for functionals of two independent Gaussian processes, including fractional, sub-fractional, and bi-fractional Brownian motions, revealing new phenomena in nonstationary cases.

Contribution

It extends limit theorems to a broader class of Gaussian processes and uncovers additional randomness in their limiting distributions.

Findings

Limit theorems apply to fractional, sub-fractional, and bi-fractional Brownian motions.

Extra randomness appears in the limits for nonstationary Gaussian processes.

Method of moments with Fourier analysis and chaining techniques is used.

Abstract

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in \cite{nx1} and a paring technique are employed.