1 results
On a class of random walks in simplexes
- Part of
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- Journal:
- Journal of Applied Probability / Volume 57 / Issue 2 / June 2020
- Published online by Cambridge University Press:
- 16 July 2020, pp. 409-428
- Print publication:
- June 2020
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- Article
- Export citation
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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.
