Two Characterizations of the Dirichlet Distribution

Abstract

Let $X = (X_1, \cdots, X_k)$ be a random vector with all $X_i \geqq 0$ and $\sum X_i \leqq 1$. Let $k \geqq 2$, and suppose that none of the $X_i$, nor $1 - \sum X_i$ vanishes almost surely. Without any further regularity assumptions, each of two conditions is shown to be necessary and sufficient for $X$ to be distributed according to a Dirichlet distribution or a limit of such distributions. Either condition requires that certain proportions between components of $X$ be independent of one or more other components of $X$.