A Pearson-Dirichlet random walk

TL;DR

This paper introduces a generalized class of Pearson-Dirichlet random walks with step lengths following a Dirichlet distribution, deriving their endpoint distributions and exploring their properties in various dimensions.

Contribution

It extends previous models by generalizing step lengths to Dirichlet distributions and characterizes the joint endpoint distributions in multiple dimensions.

Findings

Endpoint distribution inside a ball for specific steps and dimensions

Existence of two families of Pearson-Dirichlet walks with shared properties

New walks with uniform endpoint distribution derived from integral formulas

Abstract

A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1, 2, 4 . Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which are distributed over the unit (n-1) simplex according to a Dirichlet distribution whose parameters are all equal to q, a given positive value. The walk and the flight above correspond to q=1. For any d >= 3, there exist, for integer and half-integer values of q, two families of Pearson-Dirichlet walks which share a common property. For any n, the d components of the endpoint are jointly distributed as are the d components of a vector uniformly distributed over the surface of a hypersphere of radius l in a space Rk whose dimension k is an affine function of n for a given d. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in R3 and two walks in R4. Pearson-Liouville random walks, obtained by distributing the total lengths of the previous Pearson-Dirichlet walks, are finally discussed.