Decomposition and limit theorems for a class of self-similar Gaussian processes

TL;DR

This paper introduces a new class of self-similar Gaussian processes combining fractional Brownian motion and another Gaussian process, providing decomposition, limit theorems, and connections to stochastic heat equations.

Contribution

It defines a novel class of self-similar Gaussian processes, proves their decomposition into known processes, and establishes a central limit theorem for their Hermite variations.

Findings

Process decomposes into fractional Brownian motion and a Gaussian process.

Established a central limit theorem for Hermite variations.

Connected the processes to solutions of fractional-colored stochastic heat equations.

Abstract

We introduce a new class of self-similar Gaussian stochastic processes, where the covariance is defined in terms of a fractional Brownian motion and another Gaussian process. A special case is the solution in time to the fractional-colored stochastic heat equation described in Tudor (2013). We prove that the process can be decomposed into a fractional Brownian motion (with a different parameter than the one that defines the covariance), and a Gaussian process first described in Lei and Nualart (2008). The component processes can be expressed as stochastic integrals with respect to the Brownian sheet. We then prove a central limit theorem about the Hermite variations of the process.