  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.
