On spectral properties of stationary random processes connected by a special random time change

Yu. V. Yakubovich ; O. V. Rusakov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 30, Tome 501 (2021), pp. 315-334

Résumé

We consider three independent objects: a two-sided stationary random sequence $\boldsymbol{\xi} := (\ldots, \xi_{-1}, \xi_0, \xi_{1}, \ldots)$ with zero mean and finite variance, a standard Poisson process $\Pi$ and a subordinator $S$, that is a non-decreasing Lévy process. By means of reflection about zero we extend $\Pi$ and $S$ to the negative semi-axis and define a random time change $\Pi(S(t))$, $t\in\mathbb{R}$. Then we define a so-called PSI-process $\psi(t) := \xi_{\Pi(S(t))}$, $t\in\mathbb{R}$, 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 $\psi$ in terms of spectral characteristics of the sequence $\xi$ 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 $\psi$. We also determine exact spectral characteristics of $\psi$ for the following examples of $\boldsymbol{\xi}$: 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.