Maxima of bivariate random vectors: Between independence and complete dependence
type: Item
instance of: scholarly article
title: Maxima of bivariate random vectors: Between independence and complete dependence
zbMATH Open document ID: 0811.60038
DOI: 10.1016/0167-7152(94)00036-0
published in: Statistics \& Probability Letters
publication date: 1995-01-09T00:00:00Z
review text: Let \(\{(X_{in}, Z_{in})\), \(1\leq i\leq n\), \(n\geq 1\}\) be a triangular array of bivariate random vectors which are independent for each fixed \(n\). Furthermore, let \(M_{n1}= \max_{1\leq i\leq n} X_{in}\) and \(M_{n2}= \max_{1\leq i\leq n} Z_{in}\). Suppose that there are sequences \(\{a_{ni}\}\), \(\{b_{ni}\}\) such that \(P(a_{ni} (M_{ni}- b_{ni})\leq x)\to G_ i(x)\) for \(i=1,2\). \(G_ i\) is assumed to be continuous. The author gives sufficient conditions such that \[ P(a-{n_ 1} (M_{n1}- b_{n1})\leq x,\;a_{n2} (M_{n2}- b_{n2})\leq z) @>d>> G(x,z). \tag{1} \] Among other things it is proved that if \(P(a_{n2} (Z_{1n}- b_{n2})> z\mid a_{n1} (X_{1n}- b_{n1})= x)\to 0\) \((n\to\infty)\) for every continuity point \((x,z)\), then (1) holds with \(G(x,z)= G_ 1(x) G_ 2(z)\). Moreover, he derives necessary conditions for independence and complete dependence (i.e. \(G(x,z)= \min\{ G_ 1(x), G_ 2(z)\}\) of the limit distributions. As a special model he considers the interesting case \(Z_ i= X_ i+ \rho_ n Y_ i\) with \(\rho_ n>0\). The variables \(\{X_ i\}\) and \(\{Y_ i\}\) are assumed to be i.i.d. and independent of each other. If \(F_ X\) and \(F_ Y\) belong to a domain of attraction of the Fréchet distribution, it is shown that the limit distribution \(G\) is in general not max-stable. In addition, it is described how his results can be extended to triangular arrays of random vectors which are for large \(n\) stationary.
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 706915
zbMATH Keywords: maxima
zbMATH Keywords: triangular array of bivariate random vectors
zbMATH Keywords: independence and complete dependence
zbMATH Keywords: domain of attraction of the Fréchet distribution
author: J. Hüsler
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)00036-0
OpenAlex ID: W2067074025
cites work: Max-infinite divisibility
cites work: Q5560061
cites work: Extreme values of independent stochastic processes
cites work: Extreme value theory for multivariate stationary sequences
cites work: Multivariate extreme values in stationary random sequences
cites work: Maxima of normal random vectors: Between independence and complete dependence
cites work: Extremes and related properties of random sequences and processes
cites work: Semi-min-stable processes
cites work: Approximate distributions of order statistics. With applications to nonparametric statistics
cites work: Preservation of Weak Convergence Under Mapping
description: scientific article; zbMATH DE number 706915