Extreme value distributions for random coupon collector and birthday problems
type: Item
instance of: scholarly article
title: Extreme value distributions for random coupon collector and birthday problems
zbMATH Open document ID: 1003.60052
DOI: 10.1023/A:1013921125928
author: Lars Holst
published in: Extremes
publication date: 2002-11-21T00:00:00Z
review text: For a random experiment with \(n\) outcomes having probabilities \(p_1,\dots,p_n\) independent trials are performed. \(N_n\) is the number of trials needed until each outcome has occurred at least \(c\) times, \(N^*_n\) is the number of trials when some unspecified outcome has occurred \(c\) times. The author considers the case when \(p_i\) are random and given by \(p_i=X_i/\sum_{j=1}^n X_j\), where \(X_j\) are i.i.d. Using the embedding in Poisson point process the author derives the asymptotic distributions of \(N_n\) and \(N^*_n\) for different types of \(X_j\) distributions. E.g. if \(\Pr(X_j\leq x)=x^\alpha L(x)\) (\(x\to 0\)), where \(L\) is slowly varying at 0 and \(n a_n^{-\alpha}L(1/a_n)\Gamma(\alpha+c)/(c+1)!\to 1\), then \[ \Pr(N_n/(na_n\mu)\leq y)\to \exp(-y^{-\alpha}),\quad y>0, \] where \(\mu=E(X)\).
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 1833862
zbMATH Keywords: Poisson embedding
zbMATH Keywords: point process
zbMATH Keywords: inverse Gaussian distribution
zbMATH Keywords: log-normal distribution
zbMATH Keywords: gamma distribution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1833862