Convergence to Fréchet distribution via Mallows distance
type: Item
instance of: scholarly article
title: Convergence to Fréchet distribution via Mallows distance
zbMATH Open document ID: 1452.60032
author: S. Mousavinasr
author: C. R. Gonçalves
author: Chang C. Y. Dorea
published in: Statistics \& Probability Letters
publication date: 2020-10-12T00:00:00Z
review text: For a sequence of random variables \(X_1,X_2,\ldots\) and a fixed \(\alpha>0\), let \(M_n=n^{-1/\alpha}\max\{X_1,\ldots,X_n\}\) have distribution function \(F_{M_n}\). Let \(Y_1,Y_2,\ldots\) be a sequence of IID Fréchet random variables, each with the distribution function \(\Phi_\alpha(x)=\exp(-x^{-\alpha})\) for \(x>0\). The authors show that if \[ \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left(|X_j-Y_j|^\alpha 1_{\{|X_j-Y_j|>bn^{1/\alpha}\}}\right)\rightarrow0 \] as \(n\rightarrow\infty\) for any \(b>0\), then the Mallows (or Wasserstein) distance \(d_\alpha(F_{M_n},\Phi_\alpha)\rightarrow0\) as \(n\rightarrow\infty\), where \(d_\alpha\) is defined for any distribution functions \(F\) and \(G\) by \[ d_\alpha(F,G)=\inf_{(X,Y)}\left\{\mathbb{E}|X-Y|^\alpha\right\}^{1/\alpha}\,, \] where \(X\) has distribution \(F\), \(Y\) has distribution \(G\), and the infimum is taken over all couplings of \(X\) and \(Y\). This convergence in Mallows distance implies convergence in distribution of \(F_{M_n}\) to \(\Phi_\alpha\). In particular, in the special case where \(X_1,X_2,\ldots\) are IID with common distribution function \(F\), this shows that if \(d_\alpha(F,\Phi_\alpha)<\infty\) then \(d_\alpha(F_{M_n},\Phi_\alpha)\rightarrow0\) as \(n\rightarrow\infty\). Results on convergence of moments of \(M_n\) to those of the Fréchet distribution also follow from this general result on convergence of Mallows distance.
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 7259028
zbMATH Keywords: Mallows distance
zbMATH Keywords: Wasserstein distance
zbMATH Keywords: Fréchet distribution
zbMATH Keywords: extremes
MaRDI profile type: Publication
full work available at URL: j.spl.2020.108776
OpenAlex ID: W3014641966
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DOI: 10.1016/J.SPL.2020.108776
description: scientific article; zbMATH DE number 7259028