On the existence of Hilbert valued periodically correlated‎ autoregressive processes

In this paper we provide sufficient condition for existence of a‎ ‎unique Hilbert valued (H-valued) periodically‎ ‎correlated solution to the first order autoregressive model‎ ‎Xn=ρnXn−1+Zn‎, ‎for \ n∈Z‎, ‎and‎ ‎formulate the existing solution and its autocovariance operator‎. ‎Also we specially investigate equivalent condition for the‎ ‎coordinate process ⟨Xn,v⟩‎, ‎for‎ ‎arbitrary element v in H‎, ‎to satisfy in some‎ ‎autoregressive model‎. ‎Finally‎, ‎we extend our result to the‎ ‎autoregressive process with finite order‎.