Limit Laws for Extremes of Dependent Stationary Gaussian Arrays

Abstract

In this paper we show that the componentwise maxima ofweakly dependent bivariate stationary Gaussian triangular arrays converge in distribution after normalisation to Hüsler-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 Hüsler-Reiss distribution. Another finding of our paper is that the componentwise maxima and componentwise minima remain asymptotically independent even in the settings of Hüsler 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.

arXiv commentA typo in Assumption A2 has been removed

References in corpus (2)

• Stationary max-stable fields associated to negative definite functions
• Minima and maxima of elliptical arrays and spherical processes

Cited by in corpus (10)

• On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian Processes
• Maxima of independent, non-identically distributed Gaussian vectors
• Asymptotics and statistical inferences on independent and non-identically distributed bivariate Gaussian triangular arrays
• Second-order asymptotics on distributions of maxima of bivariate elliptical arrays
• Higher-order expansions of distributions of maxima in a Hüsler-Reiss model
• Maxima and minima of independent and non-identically distributed bivariate Gaussian triangular arrays
• Maxima of Skew Elliptical Triangular Arrays
• Berman's inequality under random scaling
• Limit Laws for Maxima of Contracted Stationary Gaussian Sequences
• Maxima of a triangular array of multivariate Gaussian sequence