CONTINUOUS BREUER-MAJOR THEOREM: TIGHTNESS AND NONSTATIONARITY

• 成果类型: Article
• 署名作者: Campese, Simon; Nourdin, Ivan; Nualart, David
• 署名单位: University of Luxembourg; University of Kansas
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1357
• 发表日期: 2020
• 页码: 147-177
• 关键词: CENTRAL LIMIT-THEOREMS; intersection local time; nonlinear functionals; power variations

摘要

Let Y = (Y(t))(t >= 0) be a zero-mean Gaussian stationary process with covariance function rho : R -> R satisfying rho (0) = 1. Let f : R -> R be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of f is d >= 1. If integral(R) vertical bar rho(s)vertical bar(d) ds < infinity, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of Z(epsilon) := root epsilon integral(./epsilon)(0) f (Y(s)) ds converge to those of sigma W as epsilon -> 0, where W is a standard Brownian motion and a 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 easyto-check in practice. In contrast, our condition very simple, as it only requires that vertical bar f vertical bar(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.