Some limit theorems for walsh-harmonizable dyadic stationary sequences

This paper deals with a Walsh-harmonizable dyadic stationary sequence {X(k): k=0, 1, 2,…} which is represented as $\text{X(k)=}\text{∫}\text{0}\text{1}\text{ψ}{}_{\mathrm{k}}\text{(λ) dζ(λ)}$, 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?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.