Correlation theory of almost periodically correlated processes

Abstract

A second-order stochastic process X is called almost periodically correlated (PC) in the sense of Gladyshev if its mean function m( t) and covariance R( t + τ, t) are uniformly continuous with respect to t, τ and are almosst periodic functions of t for every τ. We show that the mean uniformly almost periodic processes discussed by Kawata are also almost PC in the sense of Gladyshev. If X is almost PC, then for each fixed τ the function R( t + τ, t) has the Fourier series R(t+τ, t) ∼ ∑ λ ϵ ∇τ a(λ, τ) exp(iλt) and a(λ τ) = lim A→∞ 1 2A ∫ A+C −A+C R(t+τ, t) exp(−iλt) dt exists for every λ and τ, independently of the constant c. Assuming only that a(λ, τ) exists in this sense for every λ and τ, we show a(λ, τ) is a Fourier transform a(λ τ) = ∫ R exp(iyτ) r lambda;(dy) if and only if a(0, τ) is continuous at τ = 0; under this same condition, the set Λ = {λ: a(λ, τ) ≠ 0 for some τ} is countable. We show that a strongly harmonizable process is almost PC if and only if its spectral measure is concentrated on a countable set of diagonal lines S λ = {( γ 1, γ 2): γ 2 = γ 1 − λ}; further, one may identify the spectral measure on the k th line with the measure r λ k appearing in (iii). Finally we observe that almost PC processes are asymptotically stationary and give conditions under which strongly harmonizable almost PC processes may be made stationary by an independent random time shift.