Large deviations for moving average processes

Let Z = {hellip;, − 1, 0, 1, …}, ξ, ξn, n ϵ Z a doubly infinite sequence of i.i.d. random variables in a separable Banach space B, and an, n ϵ Z, a doubly infinite sequence of real numbers with 0 ≠ ∑n ϵ z|an| < ∞. Set Xn = ∑iϵzaiξi + n, n ⩾ 1. In this article, we prove that (X1 + X2 + … + Xn)n, n ⩾ 1 satisfies the upper bound of the large deviation principle if and only if E exp qk(ξ) < ∞, for some compact subset K of B, where qk(·) is the Minkowski functional of the set K. Interestingly enough, however, the lower bound holds without any conditions at all! We will also present an asymptotic property of the corresponding rate function.