The functional Breuer-Major theorem

Abstract

Let $X=\{ X_n\}_{n\in \mathbb{Z}}$ be zero-mean stationary Gaussian sequence of random variables with covariance function $\rho$ satisfying $\rho(0)=1$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function such that $E[\varphi(X_0)^2]<\infty$ and assume that $\varphi$ has Hermite rank $d \geq 1$. The celebrated Breuer-Major theorem asserts that, if $\sum_{r\in\mathbb{Z}} |\rho(r)|^d<\infty$ then the finite dimensional distributions of $\frac1{\sqrt{n}}\sum_{i=0}^{\lfloor n\cdot\rfloor-1} \varphi(X_i)$ 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 ${\bf D}([0,1])$ of c\`adl\`ag 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 $E[|\varphi(X_0)|^{p}]<\infty$ for some $p>2$.