Poisson-process limit laws yield Gumbel max-min and min-max

  • “A chain is only as strong as its weakest link” says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j)entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the“A chain is only as strong as its weakest link” says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j)entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the min-max. The limit laws hold whenever the entries' distribution has a density, and yield highly applicable approximation tools and design tools for the max-min and min-max of large random matrices. A brief of the results presented herein is given in: Gumbel central limit theorem for max-min and min-maxshow moreshow less

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Metadaten
Author:Iddo EliazarORCiD, Ralf MetzlerORCiDGND, Shlomi ReuveniORCiD
DOI:https://doi.org/10.1103/PhysRevE.100.022129
ISSN:2470-0045
ISSN:2470-0053
Pubmed Id:http://www.ncbi.nlm.nih.gov/pubmed?term=31574727
Parent Title (English):Physical review : E, Statistical, nonlinear and soft matter physics
Publisher:American Physical Society
Place of publication:College Park
Document Type:Article
Language:English
Year of first Publication:2019
Year of Completion:2019
Release Date:2020/11/20
Volume:100
Issue:2
Page Number:12
Funder:Deutsche ForschungsgemeinschaftGerman Research Foundation (DFG) [ME 1535/7-1]; Foundation for Polish Science within an Alexander von Humboldt Polish Honorary Research Scholarship; Azrieli Foundation; Sackler Center for Computational Molecular and Materials Science
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Peer Review:Referiert
Publication Way:Open Access
Open Access / Green Open-Access