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 $Z_s(t) = \frac{1}{(2s)^{n/2}} \int_{[-st^{1/n},st^{1/n}]^n} \Big[G(\xi(x)) - \mathbb{E}[G(\xi(x))]\Big]dx$ 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 $Z_s(1)$.