The functional Breuer-Major theorem

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Nualart, David
• 署名单位: University of Luxembourg; University of Kansas
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00917-1
• 发表日期: 2020
• 页码: 203-218
• 关键词: CENTRAL LIMIT-THEOREMS

摘要

Let X = {X-n}(n is an element of Z) be zero-mean stationary Gaussian sequence of random variables with covariance function rho satisfying rho(0) = 1. Let phi : R -> R be a function such that E[phi(X-0)(2)] < infinity and assume that phi has Hermite rank d >= 1. The celebrated Breuer-Major theorem asserts that, if Sigma(r is an element of Z) vertical bar rho(r)vertical bar(d) < infinity then the finite dimensional distributions of 1/root n Sigma(left perpendicularn.right perpenducular-1)(i=0) phi(X-i) converge to those of sigma W, where W is a standard Brownian motion and sigma is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the space D([0, 1]) of cadlag functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornstein-Uhlenbeck semigroup, we show that tightness holds under the sufficient (and almost necessary) natural condition that E[vertical bar phi(X-0)vertical bar(p)] < infinity for some p > 2.