Dirichlet random walks of two steps

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.

💡 Research Summary

The paper investigates random walks consisting of exactly two steps in a d‑dimensional Euclidean space (d > 1) where the directions of the steps are independent and uniformly distributed on the unit sphere. The novelty lies in the treatment of the step lengths: they are drawn from a Dirichlet distribution, which for two steps reduces to a (possibly asymmetric) beta distribution. Because the Dirichlet law forces the sum of the step lengths to be unity, the walk is confined to the unit simplex and the endpoint lies inside the unit ball.

The authors first consider the symmetric case, i.e. a Dirichlet(q,q) law, which is equivalent to a beta distribution with identical shape parameters q > 0. Denoting the step lengths by ℓ₁ and ℓ₂ = 1 − ℓ₁ and the unit direction vectors by u₁ and u₂, the endpoint is X = ℓ₁u₁ + ℓ₂u₂. The distance from the origin, R = ‖X‖, satisfies
R² = ℓ₁² + ℓ₂² + 2ℓ₁ℓ₂ (u₁·u₂).
Since u₁·u₂ is the cosine of the angle between two independent uniformly distributed directions, its distribution is known and depends only on the dimension d. By integrating over the angular variable and then over ℓ₁ with its beta density, the authors derive a closed‑form expression for the probability density function (pdf) of R:

f_R(r) =