Schl\"omilch integrals and probability distributions on the simplex

TL;DR

This paper reviews the Schl"omilch integral and introduces a unifying probability distribution on the simplex, providing tractable moments and covariances, and defining symmetric polynomials of fractional degree.

Contribution

It presents a new distribution unifying Dirichlet generalizations, with explicit moments and covariances, and a novel integral representation for symmetric polynomials.

Findings

Derived moments and covariances for the distribution

Unified several Dirichlet generalizations into one framework

Introduced a new integral representation for symmetric polynomials

Abstract

The Schl\"omilch integral, a generalization of the Dirichlet integral on the simplex, and related probability distributions are reviewed. A distribution that unifies several generalizations of the Dirichlet distribution is presented, with special cases including the scaled Dirichlet distribution and certain Dirichlet mixture distributions. Moments and log-ratio covariances are found, where tractable. The normalization of the distribution motivates a definition, in terms of a simplex integral representation, of complete homogeneous symmetric polynomials of fractional degree.