  A n-step Pearson-Gamma random walk in Rd 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 Rd are obtained from the latter walks by
imposing that the sum of the step lengths is equal to a fixed value. When the
latter 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. The density of the endpoint position of a n- step planar walk of
this type (n\geq2), 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
(Stochastics, 82: 201, 2010). The previous result is generalized to any walk
space dimension and any number of steps n\geq2 when the parameter of the
Pearson-Dirichlet random walk is q=d>1. The endpoint density 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 depend both on d and n and Bessel numbers independent
of d.
