TY - JOUR

T1 - On spectral properties of stationary random processes connected by a special random time change

AU - Якубович, Юрий Владимирович

AU - Русаков, Олег Витальевич

PY - 2023/6/30

Y1 - 2023/6/30

N2 - We consider three independent objects: a two-sided wide-sense stationary random sequence ξ := (.. , ξ−1, ξ0, ξ1,..) 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) := ξΠ(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 L2-theory of stationary processes is used.

AB - We consider three independent objects: a two-sided wide-sense stationary random sequence ξ := (.. , ξ−1, ξ0, ξ1,..) 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) := ξΠ(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 L2-theory of stationary processes is used.

UR - https://www.mendeley.com/catalogue/7e75aa6e-b59e-3d1d-a6e5-66a85c8717c3/

U2 - 10.1007/s10958-023-06548-1

DO - 10.1007/s10958-023-06548-1

M3 - Article

VL - 273

SP - 871

EP - 883

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -