Gamma variates of fractional shape as functioials of a homogeneous multidimensional poisson process

Abstract

If events are scattered in Rⁿ in accordance with a homogeneous Poisson process and if X is the location of the event with minimal [d]l^P norm, then in the case p = n the n^th absolute powers of the coordinates of X form a sample of size n from a gamma distribution with shape parameter 1/n. In an age of parallel computing, this fact may lead to some attractive simulation methods. One possibility is to generate R = [d]X[d] and U = Y/[d]X[d] independently, perhaps by setting U = Y/[d]Y[d] where Y has any p.d.f. which is a function only of ¦Y¦. We consider for example Y having the uniform distribution in an l^P ball.

Keywords:

• Box-Muller
• Minkowski norms
• simulation