A new family of solvable Pearson-Dirichlet random walks

TL;DR

This paper introduces a new family of solvable Pearson-Dirichlet random walks in multidimensional spaces, providing explicit endpoint density formulas as mixtures of simple functions, generalizing previous planar results.

Contribution

It generalizes the endpoint density results of Pearson-Dirichlet walks to any dimension and number of steps, with explicit formulas as mixtures involving hypergeometric functions.

Findings

Endpoint density expressed as weighted mixtures of simple densities

Explicit formulas involving hypergeometric functions

Generalization from planar to multidimensional 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.