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A Characterisation of Transient Random Walks on Stochastic Matrices with Dirichlet Distributed Limits

Part of: Limit theorems Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  19 February 2016

S. McKinlay
S. McKinlay*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Australia. Email address: s.mckinlay@ms.unimelb.edu.au.
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Abstract

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We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n − 1) · · · X(1) of independent and identically distributed copies X(k) of X converge almost surely as n → ∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

Keywords

Products of random matricesDirichlet distributionlimit distributionMarkov chainrandom exchange modelrandom nested simplicesservice networks with polling

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 60F99: None of the above, but in this section 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 542 - 555
Copyright
© Applied Probability Trust 

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