Some limit theorems for walsh-harmonizable dyadic stationary sequences

Abstract

This paper deals with a Walsh-harmonizable dyadic stationary sequence { X( k): k=0, 1, 2,…} which is represented as X(k)= ∫ 0 1 ψ k(λ) dζ(λ) , where ψ k ( λ) is the k-th Walsh function and ζ( λ) is a second-order process with orthogonal increments. One of the aims is to express the process { ζ( λ): λ ϵ[0, 1)} in terms of the Walsh–Stieltjes series ∑ X( k) ψ k ( λ) of the original sequence X( k). In order to do this a Littlewood's Tauberian theorem for a series of random variables is introduced. A finite Walsh series expression of X( k) is derived by introducing an approximate Walsh series of X( k). Also derived is a strong law of large numbers for the dyadic stationary sequences.