Continuous Breuer-Major theorem for vector valued fields

Abstract

Let be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function and let 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 then the finite dimensional distributions of converge to that of a scaled Brownian motion as Here we give a proof for the case when is a random vector field. We also give a proof for the functional convergence in of Z(s) to hold under the condition that for some p > 2, where gamma(m) denotes the standard Gaussian measure on and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of Z(s)(1).