Linear de-preferential urn models

In this paper we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the de-preferential urn scheme. We establish the almost-sure limit of the random config...

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Bibliographic Details
Published inAdvances in applied probability Vol. 50; no. 4; pp. 1176 - 1192
Main Authors Bandyopadhyay, Antar, Kaur, Gursharn
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2018
Applied Probability Trust
Subjects
Central limit theorem
Color
Configurations
Mathematical analysis
Matrix methods
Original Article
Probability
Weighting functions
Primary 60F05
Central limit theorem
urn model
de-preferential urn
strong law of large numbers
60F15
Secondary 60G57
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ISSN0001-8678
1475-6064
DOI10.1017/apr.2018.55

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Summary:In this paper we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. We refer to it as the de-preferential urn scheme. We establish the almost-sure limit of the random configuration for any balanced replacement matrix R. In particular, we show that the limiting configuration is uniform on the set of colours if and only if R is a doubly stochastic matrix. We further establish the almost-sure limit of the vector of colour counts and prove central limit theorems for the random configuration as well as for the colour counts.
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ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2018.55