Two-step Dirichlet random walks

• Pdf’s of endpoints of symmetric two-step Dirichlet random walks in RdRd.
• Role of permutation invariant pdf’s linked to asymmetric step length distributions.
• Pdf’s of endpoints of asymmetric two-step Dirichlet random walks in RdRd.

Random walks of nn steps taken into independent uniformly random directions in a dd-dimensional Euclidean space (d⩾2)(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 nn positive parameters, generalizes the beta distribution (n=2)(n=2). It is simply obtained from nn 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 qq 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 qq and for all d⩾2d⩾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 qq and q+sq+s where ss is a positive integer.