Extreme value theory for moving average processes with light-tailed innovations

Abstract

We consider stationary infinite moving average processes of the form $Y_n = \sum c_i Z_{n+i}$, where the sum ranges over the integers, (Z_i) is a sequence of iid random variables with ``light tails'' and (c_i) is a sequence of positive and summable coefficients. By light tails we mean that Z_0 has a bounded density $f(t)$ behaving asymptotically like $v(t) \exp (-g(t) )$, where v(t) behaves roughly like a constant as t goes to infinity, and g(t) is strictly convex satisfying certain asymptotic regularity conditions. We show that the iid sequence associated with Y_0 is in the maximum domain of attraction of the Gumbel distribution. Under additional regular variation conditions on g, it is shown that the stationary sequence (Y_n) has the same extremal behaviour as its associated iid sequence. This generalizes results of Rootz\'en (1986, 1987), where $g(t) = t^p$ and $v(t)=c t^d$ for p > 1, positive c and a real constant d