On a class of random walks in simplexes

We study the limit behaviour of a class of random walk models taking values in the standard d-dimensional simplex. From an interior point z, the process chooses one of the 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.