On a class of probability distributions

Abstract

1. The statement of the problem. A problem which arises in certain physical (5) and astronomical (1, 3) contexts relates to the probability distribution of the infinite sum F=Σ^∞_j=1r_j/rⁿ⁺¹_j(|r_j|=r_j), where the r_j's are the position vectors from a fixed origin of an infinite uniform Poisson distribution of points with a given constant mean space density N. While certain special cases of the sum (1) have been considered in the literature (cf. (2)) the study of the general case discloses properties of a class of probability distributions which appear to have some interest. For example, we shall show that the probability distribution of F exists only for n>3/2 that the moments of F (= |F|) exist only for orders 0 ≤ p < 3/n, and finally that an explicit formula can be given for all the finite moments.