Schur properties of convolutions of gamma random variables
type: Item
instance of: scholarly article
title: Schur properties of convolutions of gamma random variables
zbMATH Open document ID: 1355.60027
author: Farbod Roosta-Khorasani
author: G. J. Székely
published in: Metrika
publication date: 2015-11-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1601.04731
full work available at URL: 1601.04731v1.pdf
review text: The authors consider Schur convexity and concavity properties for tail probabilities of linear combinations of i.i.d. gamma random variables. That is, letting \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_n)\) and \(\boldsymbol{\mu}=(\mu_1,\ldots,\mu_n)\) be elements of \(\mathbb{R}_+^n\) such that \(\boldsymbol{\lambda}\) majorizes \(\boldsymbol{\mu}\), the authors give conditions on \(x\in\mathbb{R}_+\) under which \[ P\left(\sum_{i=1}^n\lambda_iX_i<x\right)\leq P\left(\sum_{i=1}^n\mu_iX_i<x\right), \] where \(X_1,X_2,\ldots\) are i.i.d. gamma random variables. Similar conditions are also given for the reverse inequality to hold. These results improve similar inequalities in the literature, and are also extended to the case where \(\boldsymbol{\lambda}\) only weakly majorizes \(\boldsymbol{\mu}\), and to the infinite-dimensional setting. The proofs make use of Laplace transforms. Applications to signal detection and matrix trace estimation are also discussed.
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 6505299
zbMATH Keywords: Schur convexity of tails
zbMATH Keywords: majorization order
zbMATH Keywords: gamma distribution
zbMATH Keywords: tail probabilities
MaRDI profile type: Publication
OpenAlex ID: W3103156352
arXiv ID: 1601.04731
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DOI: 10.1007/S00184-015-0537-9
description: scientific article