On a class of random walks in simplexes

TL;DR

This paper investigates the long-term behavior of specific random walks within the simplex, demonstrating convergence to Dirichlet distributions and the arcsine law in different models.

Contribution

It introduces a class of simplex-based random walks and proves their convergence to Dirichlet and arcsine distributions, extending understanding of their limiting behaviors.

Findings

Limit distributions are Dirichlet in certain cases.

A related model converges to the arcsine law.

Provides new insights into random walks in simplexes.

Abstract

We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. 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 specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, 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 the arcsine law.