Extremes for Solutions to Stochastic Difference Equations with Regularly Varying Tails

Abstract

The main purpose of this paper is to look at the extremal properties of Xk = X∞ j=1 j Y−1 s=1 Ak−s ! Bk−j , k ∈ Z , where (Ak, Bk)k∈Z is a periodic sequence of independent R 2 +-valued random pairs. The so-called complete convergence theorem we prove enable us to give in detail the weak limiting behavior of various functional of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the maximum and its corresponding extremal index. An application to a particular class of bilinear processes is included. These results generalize the ones obtained for the stationary case.