Certain Periodically Correlated Multi-component Locally Stationary Processes

TL;DR

This paper introduces a novel class of periodically correlated multi-component locally stationary processes, analyzing their covariance structure and spectral properties, with potential applications in time series modeling.

Contribution

It defines a new process combining locally stationary and periodically correlated components, and characterizes its covariance and spectral properties.

Findings

Covariance function has multi-component locally stationary form.

Existence of a bi-periodic correlation measure is established.

Spectral representation of the combined process is derived.

Abstract

By introducing $X^{ls}(t)$ as a random mixture of two stationary processes where the time dependent random weights have exponentially convex covariance, we show that this process has a multi-component locally stationary covariance function in Silverman's sense. We also define $X^p(t)$ as a certain continuous time periodically correlated (PC) process where its covariance function is generated by the covariance function of a discrete time through defining some simple random measure on real line. We also impose a bi-periodic correlation for this PC process with $X^{ls}(t)$. The existence of such random measure is proved. Then by defining $X(t)=X^{ls}(t)+X^p(t)$ as a certain periodically correlated multi-component locally stationary process, the covariance structure and time varying spectral representation of such processes are characterized.