Minima and maxima of elliptical arrays and spherical processes

Abstract

In this paper, we investigate first the asymptotics of the minima of elliptical triangular arrays. Motivated by the findings of Kabluchko (Extremes 14 (2011) 285-310), we discuss further the asymptotic behaviour of the maxima of elliptical triangular arrays with marginal distribution functions in the Gumbel or Weibull max-domain of attraction. We present an application concerning the asymptotics of the maximum and the minimum of independent spherical processes.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ463 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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

“… ≤ e −(b n /(4a n q)) 2 /2 < ϵ/n, with a n and b n defined in (16). Similarly, the second term in (27) is bounded from above by…”

“…The finite-dimensional distributions of a Brown-Resnick process can be naturally identified as the so-called Hüsler-Reiss distributions introduced in Hüsler and Reiss [22] which appear as the limits of maxima of a triangular array of Gaussian random vectors. Those distributions arise even in more general, non-Gaussian settings, as shown in Hashorva [16] and Hashorva et al [17]. In fact the latter paper provides conditions for the weak convergence of maxima of independent, multivariate chi-square random vectors to the Hüsler-Reiss distribution.…”

Higher-order expansions of distributions of maxima in a Hüsler-Reiss model

“…In fact, the bivariate Hüsler-Reiss distribution appeared in another context in [1], see for recent contribution in this direction [4,17,20,2]. Related results for more general triangular arrays can be found in [9,5,6,8,11,13,14,15,10,3,12]; an interesting statistical applications related to the Hüsler-Reiss distribution is presented in [6]. For both applications and various theoretical investigations, it is of interest to know how good the Hüsler-Reiss distribution approximates the distribution of the bivariate maxima.…”

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.…”