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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. [ABSTRACT FROM AUTHOR]

n n-step Pearson-Gamma random walk in ℝ starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ℝ are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥ d and any n≥2 when q is either $q = \frac{d}{2} - 1 $ ( d=3) or q= d−1 ( d=2) (Le Caër in J. Stat. Phys. 140:728-751, ). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type ( n≥2), with q= d=2, was shown recently to be a weighted mixture of 1+ floor( n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201-229, ). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q= d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q= d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+ floor( n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d. [ABSTRACT FROM AUTHOR]

A constrained diffusive random walk of n steps in ℝ d and a random flight in ℝ d, which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, ; J. Theor. Probab. 20:769, , and J. Stat. Phys. 131:1039, ). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which have independent and identical gamma distributions with a shape parameter q>0. Given the total walk length being equal to 1, the step lengths have a Dirichlet distribution whose parameters are all equal to q. The walk and the flight above correspond to q=1. Simple analytical expressions are obtained for any d≥2 and n≥2 for the endpoint distributions of two families of walks whose q are integers or half-integers which depend solely on d. These endpoint distributions have a simple geometrical interpretation. Expressed for a two-step planar walk whose q=1, it means that the distribution of the endpoint on a disc of radius 1 is identical to the distribution of the projection on the disc of a point M uniformly distributed over the surface of the 3D unit sphere. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in ℝ3, two of two steps and two of three steps, and one walk of two steps in ℝ4. Pearson–Liouville random walks, obtained by distributing the total lengths of the previous Pearson–Dirichlet walks according to some specified probability law are finally discussed. Examples of unconstrained random walks, whose step lengths are gamma distributed, are more particularly considered. [ABSTRACT FROM AUTHOR]