Extremes of independent chi-square random vectors

Abstract

We prove that the componentwise maximum of an i.i.d. triangular array of chi-square random vectors converges in distribution, under appropriate assumptions on the dependence within the vectors and after normalization, to the max-stable Hüsler-Reiss distribution. As a by-product we derive a conditional limit result.

Gaussian approximation of perturbed chi-square risks

“…The seminal contribution [26] shows that for bivariate Gaussian triangular arrays (X 2 ) as defined in (2), where in (1) we put ρ n ∈ (−1, 1)/{0} instead of ρ. In [21] the result of [26] was extended to chi-square case proving that under the condition (24) (set below t n (x) = a n x + b n ) lim n→∞ sup x,y∈R (H n (t n (x), t n (y))) n − H λ (x, y) = 0,…”

Maxima of independent, non-identically distributed Gaussian vectors

“…The generalization to multivariate elliptical distributions can be found in Hashorva [14]. Moreover, Hashorva et al [15] prove that also some non-elliptical distributions are in the domain of attraction of the Hüsler-Reiss distribution, for instance multivariate χ 2 -distributions.…”

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.…”