Continuous Breuer-Major theorem for vector valued fields

Abstract

Let ξ : Ω × R n → R be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function r(x) = E[ξ(0)ξ(x)] and let G : R → 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 ∈ L d (R n ), then the finite dimensional distributions of G(ξ(x))] dx converge to that of a scaled Brownian motion as s → ∞. Here we give a proof for the case when ξ : Ω × R n → R m is a random vector field. We also give a proof for the functional convergence in C([0, ∞)) of Zs to hold under the condition that for some p > 2, G ∈ L p (R m , γm) where γm denotes the standard Gaussian measure on R m and we derive expressions for the asymptotic variance of the second chaos component in the Wiener chaos expansion of Zs(1).2010 Mathematics Subject Classification. 60F05, 60F17, 60G15, 60G60, 60H07.