Limit laws for extremes of dependent stationary Gaussian arrays

Abstract

In this paper we show that the componentwise maxima of weakly 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.

Maxima of a triangular array of multivariate Gaussian sequence

“…By letting ρ depend on the sample size and go to one with certain rate, Hüsler and Reiss (1989) showed that the normalized maxima can become asymptotically dependent. In this paper, we extend such a study to a triangular array of multivariate Gaussian sequence, which further generalizes the results in Hsing, Hüsler and Reiss (1996) and Hashorva and Weng (2013). …”

“…Such a study will generalize the results in both Hsing, Hüsler and Reiss (1996) and Hashorva and Weng (2013).…”

“…Another extension of Hüsler and Reiss (1989) made by Hashorva and Weng (2013) is to study a triangular array of 2-dimensional stationary Gaussian sequence as follows.…”

Extremal behavior of squared Bessel processes attracted by the Brown–Resnick process

“…A natural extension could be thus to consider maxima of non-identically distributed independent Gaussian processes and their functional limits. Furthermore, the independence assumption can be eventually relaxed as in Hashorva and Weng [19], so that the limit process still remains Brown-Resnick. With regard to applications, there have been some developments in simulating Brown-Resnick processes [9,12,27].…”

The joint distribution of the maximum and minimum of an AR(1) process

“…However, the recent years have seen considerable developments on the asymptotics of the joint distribution of M n and m n following on from Davis (1979). We mention: the joint distribution of M n and m n for complete and incomplete samples of stationary sequences (Peng et al, 2010); the joint distribution of M n and m n for complete and incomplete samples from weakly dependent stationary sequences (Peng et al, 2011); the joint distribution of M n and m n for strongly dependent Gaussian vector sequences (Weng et al, 2012); the joint distribution of M n and m n for independent spherical processes (Hashorva, 2013); the joint distribution of M n and m n for dependent stationary Gaussian arrays (Hashorva and Weng, 2013); large deviation results on M n and m n for independent and identical samples (Giuliano and Macci, 2014); the joint distribution of M n and m n for scaled stationary Gaussian sequences (Hashorva et al, 2014); the joint distribution of M n and m n for complete and incomplete stationary sequences (Hashorva and Weng, 2014a,b); the joint distribution of M n and m n for Hüsler-Reiss bivariate Gaussian arrays (Liao and Peng, 2014). None of these papers give the exact joint distribution of M n and m n for finite n.…”