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Linear de-preferential urn models

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  29 November 2018

Antar Bandyopadhyay  and
Gursharn Kaur
Antar Bandyopadhyay*
Affiliation:
Indian Statistical Institute
Gursharn Kaur*
Affiliation:
Indian Statistical Institute
*
* Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi Centre, 7 SJS Sansanwal Marg, New Delhi 110016, India.
* Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi Centre, 7 SJS Sansanwal Marg, New Delhi 110016, India.
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Abstract

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.

Keywords

Central limit theoremde-preferential urnstrong law of large numbersurn model

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems
Secondary: 60G57: Random measures
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1176 - 1192
Copyright
Copyright © Applied Probability Trust 2018 

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