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

Abstract

We consider stationary infinite moving average processes of the form $$Y_n=\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_i{Z}_{n+i},n\in \mathbb{Z},$$ where (Z_i)_i∈Z is a sequence of independent and identically distributed (i.i.d.) random variables with light tails and (c_i)_i∈Z is a sequence of positive and summable coefficients. By `light tails' we mean that Z₀ has a bounded density $f\left(t\right)\sim \nu \left(t\right)exp(-\psi (t\left)\right)$ , 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 (Y_n)_n∈N has the same extremal behaviour as its associated i.i.d. sequence. This generalizes Rootzén's results where $f\left(t\right)\sim c{t}^{\alpha }exp(-t^p)\mathrm{for}c>0,\alpha \in R\mathrm{and}p>1$