Dirichlet random walks of two steps

TL;DR

This paper derives explicit probability density functions for the endpoint distance of two-step Dirichlet random walks in higher dimensions, generalizing previous models and including symmetric and asymmetric cases.

Contribution

It provides a new explicit formula for the endpoint distribution of symmetric two-step Dirichlet walks in any dimension, extending to asymmetric beta-distributed step lengths.

Findings

Explicit density function for symmetric Dirichlet two-step walk in any dimension.

Density expression for asymmetric beta-distributed two-step walk.

Generalization of Dirichlet walk models to asymmetric cases.

Abstract

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d larger than 1), are named Dirichlet when their step lengths are distributed according to a Dirichlet law. The latter continuous multivariate distribution, which depends on n positive parameters, generalizes the beta distribution (n=2). The sum of step lengths is thus fixed and equal to 1. In the present work, the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric Dirichlet random walk of two steps which depends on a single positive parameter q. It is valid for any positive q and for all d larger than 1. The latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters, namely q and q+s where s is a positive integer.