Dirichlet random walks

Abstract

This article provides tools for the study of the Dirichlet random walk in $\mathbb{R}^d$. By this we mean the random variable $W=X_1\Theta_1+\cdots+X_n\Theta_n$ where $X=(X_1,\ldots,X_n) \sim \mathcal{D}(q_1,\ldots,q_n)$ is Dirichlet distributed and where $\Theta_1,\ldots \Theta_n$ are iid, uniformly distributed on the unit sphere of $\mathbb{R}^d$ and independent of $X.$ In particular we compute explicitely in a number of cases the distribution of $W.$ Some of our results appear already in the literature, in particular in the papers by G\'erard Le Ca\"{e}r (2010, 2011). In these cases, our proofs are much simpler from the original ones, since we use a kind of Stieltjes transform of $W$ instead of the Laplace transform: as a consequence the hypergeometric functions replace the Bessel functions. A crucial ingredient is a particular case of the classical and non trivial identity, true for $0\leq u\leq 1/2$:$$_2F_1(2a,2b;a+b+\frac{1}{2};u)= \_2F_1(a,b;a+b+\frac{1}{2};4u-4u^2).$$ We extend these results to a study of the limits of the Dirichlet random walks when the number of added terms goes to infinity, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and of Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit ball of $\mathbb{R}^d.$ {4mm}\noindent \textsc{Keywords:} Dirichlet processes, Stieltjes transforms, random flight, distributions in a ball, hyperuniformity, infinite divisibility in the sense of Dirichlet. {4mm}\noindent \textsc{AMS classification}: 60D99, 60F99.