Continuous Breuer-Major theorem for vector valued fields. Issue 4 (3rd July 2020)

• Record Type: Journal Article
• Authors: Nualart, David; Tilva, Abhishek
• Is Part Of: Stochastic analysis and applications. Volume 38:Issue 4(2020)
• Page Start: 668
• Page End: 685
• Publication Date: 2020-07-03
• Subjects: Breuer-Major theorem -- functional limit theorem -- Wiener chaos expansions; 60F05 -- 60F17 -- 60G15 -- 60G60 -- 60H07; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2019.1711118
• Languages: English
• ISSNs: 0736-2994

Abstract

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 Z s ( t ) = 1 ( 2 s ) n / 2 ∫ [ − s t 1 / n, s t 1 / n ] n [ G ( ξ ( x ) ) − E [ G ( ξ ( x ) ) ] ] d x 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, γ m ) 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 Z s ( 1 ) .