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Article Dans Une Revue Physica A: Statistical Mechanics and its Applications Année : 2015

Two-step Dirichlet random walks

Gérard Le Caër

Résumé

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d>1) , 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>1 . The latter pdf is used in turn to express the related density of a random walk of two steps whose 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.
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Dates et versions

hal-01132185 , version 1 (18-03-2015)
hal-01132185 , version 2 (24-03-2015)

Identifiants

Citer

Gérard Le Caër. Two-step Dirichlet random walks. Physica A: Statistical Mechanics and its Applications, 2015, 430, pp.201-215. ⟨10.1016/j.physa.2015.02.075⟩. ⟨hal-01132185v2⟩
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