caa a22 4500 445390948 CHVBK 20180317143046.0 cr unu---uuuuu 170323e20110701xx s 000 0 eng 10.1007/s10955-011-0245-4 doi (NATIONALLICENCE)springer-10.1007/s10955-011-0245-4 Le Caër Gérard Institut de Physique de Rennes, UMR UR1-CNRS 6251, Université de Rennes I, Campus de Beaulieu, Bâtiment 11A, 35042, Rennes Cedex, France aut A New Family of Solvable Pearson-Dirichlet Random Walks [Elektronische Daten] [Gérard Le Caër] An n-step Pearson-Gamma random walk in ℝ d 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 d 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 0 and any n≥2 when q is either $q = \frac{d}{2} - 1 $ (d 0=3) or q=d−1 (d 0=2) (Le Caër in J. Stat. Phys. 140:728-751, 2010). 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, 2010). 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. Springer Science+Business Media, LLC, 2011 Pearson-Rayleigh random walks nationallicence Random flights nationallicence Dirichlet distribution nationallicence Gamma distribution nationallicence Gauss hypergeometric function nationallicence Bessel numbers nationallicence Inverse Laplace transform nationallicence Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/1(2011-07-01), 23-45 0022-4715 144:1<23 2011 144 10955 https://doi.org/10.1007/s10955-011-0245-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-011-0245-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Le Caër Gérard Institut de Physique de Rennes, UMR UR1-CNRS 6251, Université de Rennes I, Campus de Beaulieu, Bâtiment 11A, 35042, Rennes Cedex, France aut NATIONALLICENCE 773 0- Journal of Statistical Physics Springer US; http://www.springer-ny.com 144/1(2011-07-01), 23-45 0022-4715 144:1<23 2011 144 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer