On Spectral Properties of Stationary Random Processes Connected by a Special Random Time Change

We consider three independent objects: a two-sided wide-sense stationary random sequence ξ := (. . . , ξ −1 , ξ 0 , ξ 1 , . . . ) with zero mean and finite variance, a standard Poisson process Π and a subordinator S, that is a nondecreasing Lévy process. By means of reflection about zero we extend Π and S to the negative semi-axis and define a random time change Π(S(t)), t ∈ ℝ. Then we define a so-called PSI-process ψ(t) := ξ Π(S(t)) , t ∈ ℝ, which is wide-sense stationary. Notice that PSI-processes generalize pseudo-Poisson processes. The main aim of the paper is to express spectral properties of the process ψ in terms of spectral characteristics of the sequence ξ and the Lévy measure of the subordinator S. Using complex analytic techniques, we derive a general formula for the spectral measure G of the process ψ. We also determine exact spectral characteristics of ψ for the following examples of ξ: almost periodic sequence; finite-order moving average; finite order autoregression. These results can find their applications in all areas where L 2 -theory of stationary processes is used.