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

Abstract

We consider stationary infinite moving average processes of the form Y n =∑ i =-∞ ∞c iZ n +i,n∈ℤ, where (Zi)i∈Z is a sequence of independent and identically distributed (i.i.d.) random variables with light tails and (ci)i∈Z is a sequence of positive and summable coefficients. By `light tails' we mean that Z0 has a bounded density f (t)∼ν(t)exp(-ψ(t)) , where ν(t) behaves roughly like a constant as t →∞ and ψ is strictly convex satisfying certain asymptotic regularity conditions. We show that the i.i.d. sequence associated with Y0 is in the maximum domain of attraction of the Gumbel distribution. Under additional regular variation conditions on ψ, it is shown that the stationary sequence (Yn)n∈N has the same extremal behaviour as its associated i.i.d. sequence. This generalizes Rootzén's results where f (t)∼ct αexp(-t p )forc>0,α∈Randp>1