Limit Laws for Extremes of Dependent Stationary Gaussian Arrays

TL;DR

This paper investigates the asymptotic behavior of componentwise maxima in weakly dependent stationary Gaussian arrays, revealing convergence to Hüsler-Reiss distribution and independence properties under various dependence conditions.

Contribution

It establishes new limit laws for maxima of dependent Gaussian arrays, including convergence to Hüsler-Reiss distribution and asymptotic independence of maxima and minima.

Findings

Maxima converge to Hüsler-Reiss distribution under weak dependence.

Maxima and minima remain asymptotically independent.

Almost sure limit theorem derived under Berman condition.

Abstract

In this paper we show that the componentwise maxima ofweakly dependent bivariate stationary Gaussian triangular arrays converge in distribution after normalisation to H\"usler-Reiss distribution. Under a strong dependence assumption, we prove that the limit distribution of the maxima is a mixture of a bivariate Gaussian distribution and H\"usler-Reiss distribution. Another finding of our paper is that the componentwise maxima and componentwise minima remain asymptotically independent even in the settings of H\"usler and Reiss (1989) allowing further for weak dependence. Further we derive an almost sure limit theorem under the Berman condition for the components of the triangular array.