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January 2020 Continuous Breuer–Major theorem: Tightness and nonstationarity
Simon Campese, Ivan Nourdin, David Nualart
Ann. Probab. 48(1): 147-177 (January 2020). DOI: 10.1214/19-AOP1357

Abstract

Let $Y=(Y(t))_{t\geq 0}$ 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)|^{d}\,ds<\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. First, 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 [J. Multivariate Anal. 80 (2002) 191–216] 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.

Second, 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 Brownian motion concludes the paper.

Citation

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Simon Campese. Ivan Nourdin. David Nualart. "Continuous Breuer–Major theorem: Tightness and nonstationarity." Ann. Probab. 48 (1) 147 - 177, January 2020. https://doi.org/10.1214/19-AOP1357

Information

Received: 1 July 2018; Revised: 1 March 2019; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206755
MathSciNet: MR4079433
Digital Object Identifier: 10.1214/19-AOP1357

Subjects:
Primary: 60F17, 60G15, 60H07
Secondary: 60G05

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • January 2020
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