First order autoregressive periodically correlated model in Banach spaces: Existence and central limit theorem

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.