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On a class of random walks in simplexes

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  16 July 2020

Tuan-Minh Nguyen  and
Stanislav Volkov
Tuan-Minh Nguyen*
Affiliation:
Monash University
Stanislav Volkov*
Affiliation:
Lund University
*
*Postal address: School of Mathematical Sciences, Monash University, Victoria 3800, Australia
**Postal address: Centre for Mathematical Sciences, Lund University, Lund 22100-118, Sweden
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Abstract

We study the limit behaviour of a class of random walk models taking values in the standard d-dimensional ( $d\ge 1$ ) simplex. From an interior point z, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on z, and then the particle randomly jumps to a new location z′ on the segment connecting z to the chosen vertex. In some special cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are Dirichlet. We also consider a related history-dependent random walk model in [0, 1] based on an urn-type scheme. We show that this random walk converges in distribution to an arcsine random variable.

Keywords

Random walks in simplexesiterated random functionsDirichlet distributionstick-breaking process

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 409 - 428
Copyright
© Applied Probability Trust 2020

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