Certain Periodically Correlated Multicomponent Locally Stationary Processes

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 multicomponent 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 a real line. We also impose a biperiodic correlation for this PC process with $X^{ls}(t)$. The existence of such a random measure is proved. Then by defining $X(t)=X^{ls}(t)+X^p(t)$ as a certain PC multicomponent locally stationary process, the covariance structure and time varying spectral representation of such processes are characterized.