Central limit theorem for functionals of a generalized self-similar Gaussian process

TL;DR

This paper extends the central limit theorem to a broad class of self-similar Gaussian processes without stationary increments, demonstrating Gaussian convergence of functionals under specific covariance conditions.

Contribution

It proves a generalized Breuer-Major theorem for non-stationary, self-similar Gaussian processes using the Fourth Moment Theorem, with concrete examples.

Findings

Functional convergence to Gaussian distribution under covariance conditions

Validation for five non-stationary process examples

Extension of CLT to non-stationary, self-similar processes

Abstract

We consider a class of self-similar, continuous Gaussian processes that do not necessarily have stationary increments. We prove a version of the Breuer-Major theorem for this class, that is, subject to conditions on the covariance function, a generic functional of the process increments converges in law to a Gaussian random variable. The proof is based on the Fourth Moment Theorem. We give examples of five non-stationary processes that satisfy these conditions.