LARGE DEVIATIONS FOR MOVING AVERAGE PROCESSES

Let Z = {..., - 1, 0, 1, ...}, xi, xi(n), n epsilon Z a doubly infinit e sequence of i.i.d, random variables in a separable Banach space B, a nd a(n), n epsilon Z, a doubly infinite sequence of real numbers with 0 not equal Sigma(n epsilon Z)\a(n)\ < infinity. Set X(n) = Sigma(i ep silon Z)a(i) xi(i+n), n greater than or equal to 1. In this article, w e prove that (X(1) + X(2) + ... + X(n))n/n greater than or equal to 1 satisfies the upper bound of the large deviation principle if and only if E exp q(K)(xi) < infinity, for some compact subset K of B, where q (K)(.) 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 functio n.