In this talk we will discuss some results obtained by Gerard Le Caer (J Stat Phys 2010, 2011) concerning Dirichlet random walks, namely convex combinations of points on a sphere with weights having Dirichlet distribution. The proofs are based on a kind of Stieltjes transform, suited for distributions on a ball, which has special properties with regard to the Dirichlet distribution. A limiting regime is considered, from which the means of Dirichlet processes come out.