Dirichlet integrals and a theory of marginal densities in spaces of dimension N greater than or equal to 2

Abstract

A continuous, nonnegative function f(r) on (0,R), with ..integral../sub 0//sup R/ r/sup N-1/ f(r) dr suitably defined, determines a radially symmetric probability density f(..sqrt..(..sigma../sub 1//sup N/x/sub i//sup 2/)) on the sphere 0 .. p(x/sub 1/) is one--one, and the unique function f(r) with a given marginal density p(x/sub 1/) is expressible explicitly in terms of p(x/sub 1/). Necessary and sufficient conditions are obtained for a function p(x/sub 1/) to be the marginal density of some f(r) in the case of odd N, while necessary and weak sufficient conditions are given for even N. A Dirichlet-type integral is an essential tool, and its value is here derived by a simple induction on N, which avoids the combersome transformation to the N-cube usually employed in such cases.