Continuous Breuer-Major theorem for vector valued fields

Abstract

Let $\xi :\Omega \times {\mathbb{R}}^n\to \mathbb{R}$ be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function $r(x)=\mathbb{E}[\xi (0)\xi (x)]$ and let $G:\mathbb{R}\to \mathbb{R}$ such that $G$ is square integrable with respect to the standard Gaussian measure and is of Hermite rank $d$. The Breuer-Major theorem in it's continuous setting gives that, if $r\in L^d({\mathbb{R}}^n)$, then the finite dimensional distributions of $Z_s(t)=\frac{1}{(2s{)}^{n/2}}{\int }_{[-s{t}^{1/n},s{t}^{1/n}{]}^n}[G(\xi (x))-\mathbb{E}[G(\xi (x))]]dx$ converge to that of a scaled Brownian motion as $s\to \infty$. Here we give a proof for the case when $\xi :\Omega \times {\mathbb{R}}^n\to {\mathbb{R}}^m$ is a random vector field. We also give a proof for the functional convergence in $C([0,\infty ))$ of $Z_s$ to hold under the condition that for some $p>2$, $G\in L^p({\mathbb{R}}^m,{\gamma }_m)$ where ${\gamma }_m$ denotes the standard Gaussian measure on ${\mathbb{R}}^m$ and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of $Z_s(1)$.

Added David Nualart as author. Same results under weaker conditions and with new proofs. 17 pages