Two-step Dirichlet random walks

Gérard Le Caër

Physica A: Statistical Mechanics and its Applications, 2015, vol. 430, issue C, 201-215

Abstract: Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d⩾2), which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality, are named “Dirichlet” when this constraint is realized via a Dirichlet law of step lengths. The latter continuous multivariate distribution, which depends on n positive parameters, generalizes the beta distribution (n=2). It is simply obtained from n independent gamma random variables with identical scale factors. Previous literature studies of these random walks dealt with symmetric Dirichlet distributions whose parameters are all equal to a value q which takes half-integer or integer values. 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. It is valid for any positive value of q and for all d⩾2. 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.

Keywords: Dirichlet random walks; Random flights; Dirichlet distribution; Asymmetric distributions; Pochhammer symbols; Lucas coefficients (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:430:y:2015:i:c:p:201-215

DOI: 10.1016/j.physa.2015.02.075

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