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Limit Theorems for Random Triangular URN Schemes

Part of: Nonparametric inference Stochastic processes Limit theorems Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Rafik Aguech
Rafik Aguech*
Affiliation:
Faculté des Sciences de Monastir, Tunisia
*
∗Postal address: Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir et EPAM Sousse, Tunisia. Email address: rafik.aguech@ipeit.rnu.tn
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Abstract

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In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

Keywords

Multitype branching processgeneralized Pólya urnurn modelYule process

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes 60G40: Stopping times; optimal stopping problems; gambling theory 62G20: Asymptotic properties 60K05: Renewal theory 60G46: Martingales and classical analysis 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 827 - 843
Copyright
Copyright © Applied Probability Trust 2009 

References

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