Continuous Breuer-Major theorem: tightness and non-stationarity

TL;DR

This paper advances the understanding of the continuous Breuer-Major theorem by establishing a simple tightness condition and extending it to non-stationary, self-similar Gaussian processes, with applications to bifractional Brownian motion.

Contribution

It introduces a new, practical tightness criterion using Malliavin calculus and extends the theorem to non-stationary Gaussian processes.

Findings

New tightness condition based on integrability of |f|^p for p>2

Extension of the theorem to non-stationary, self-similar Gaussian processes

Application to bifractional Brownian motion fluctuations

Abstract

Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $\rho:\mathbb{R}\to\mathbb{R}$ satisfying $\rho(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_\mathbb{R} |\rho(s)|^dds<\infty$, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_\varepsilon:=\sqrt{\varepsilon}\int_0^{\cdot/\varepsilon}f(Y(s))ds$ converge to those of $\sigma W$ as $\varepsilon\to 0$, where $W$ is a standard Brownian motion and $\sigma$ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. Firstly, we investigate the tightness in the Breuer-Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [1] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^p$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Secondly, and motivated by a problem of geometrical nature, we extend the continuous Breuer-Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Browninan motion concludes the paper.