Some limit theorems for Walsh-harmonizable dyadic stationary sequences (Q1062352)

This paper deals with a Walsh-harmonizable dyadic stationary sequence \(\{\) X(k): \(k=0,1,2,...\}\) which is represented as \(X(k)=\int^{1}_{0}\psi_ k(\lambda)d\zeta (\lambda),\) where \(\psi_ k(\lambda)\) is the k-th Walsh function and \(\zeta\) (\(\lambda)\) is a second-order process with orthogonal increments. One of the aims is to express the process \(\{\) \(\zeta\) (\(\lambda)\): \(\lambda\in [0,1)\}\) in terms of the Walsh-Stieltjes series \(\sum X(k)\psi_ k(\lambda)\) 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.