Abstract
We let B be a separable Banach space, and let {Zn} be a sequence of independent and identically distributed random elements in B. Then we prove that for a given strongly periodic sequence of bounded linear operators {ρn}, the order one autoregressive system equations Xn=ρnXn−1+Zn,n in set on integers, possesses a unique almost sure strictly periodically correlated solution; under E[log+‖Z0‖]<∞, which appears to be necessary as well. We proceed on to derive the limiting distribution of ∑n=1NXn that appears to be a Gaussian distribution on B. We also provide interesting examples and observations.
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