On the existence of Hilbert valued periodically correlated‎ autoregressive processes

Document Type : Research Paper


1 Department of Mathematical‎ ‎Sciences‎, ‎Isfahan University of Technology‎, ‎Isfahan‎, ‎Iran‎.

2 Department of Mathematical Sciences‎, ‎Isfahan‎ ‎University of Technology‎, ‎Isfahan‎, ‎Iran.

3 Department of Statistics‎, ‎Faculty of Sciences‎, ‎University of‎ ‎Isfahan‎, ‎Isfahan‎, ‎Iran.


‎In this paper we provide sufficient condition for existence of a‎ ‎unique Hilbert valued ($\mathbb{H}$-valued) periodically‎ ‎correlated solution to the first order autoregressive model‎ ‎$X_{n}=\rho _{n}X_{n-1}+Z_{n}$‎, ‎for \ $n\in \mathbb{Z}$‎, ‎and‎ ‎formulate the existing solution and its autocovariance operator‎. ‎Also we specially investigate equivalent condition for the‎ ‎coordinate process $\left\langle X_{n},v\right\rangle $‎, ‎for‎ ‎arbitrary element $v$ in $\mathbb{H}$‎, ‎to satisfy in some‎ ‎autoregressive model‎. ‎Finally‎, ‎we extend our result to the‎ ‎autoregressive process with finite order‎.


Main Subjects

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