A New Family of Solvable Pearson-Dirichlet Random Walks

Abstract

A n-step Pearson-Gamma random walk in Rd starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in Rd are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. When the latter is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q. The density of the endpoint position of a n- step planar walk of this type (n\geq2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Stochastics, 82: 201, 2010). The previous result is generalized to any walk space dimension and any number of steps n\geq2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. The endpoint density is a weighted mixture of 1+floor(n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depend both on d and n and Bessel numbers independent of d.