Coordinates of Dirichlet Distribution Negatively Associated?

Question

Suppose $X$ is sampled from a symmetric Dirichlet distribution with arbitrary shape and $n$ dimensions. Equivalently we can independently sample $z_i \sim \text{Gamma}(\alpha, 1)$ and then set $x_i=\frac{z_i}{\sum z_i}$

I want to show that the coordinates $x_i$ are negatively associated. Intuition tells me that, of course they are, since occupancy numbers (balls-in-bins) are negatively associated.

I'm considering using a scaling argument: assume that each $x_i$ is rational, and then let $\lambda$ be the least common multiple of the denominators. Then we can phrase this problem as a balls-in-bins problem where we have $1/\lambda$ balls and $n$ bins.

This isn't as elegant as I'd like, and it seems that this is the type of thing results must exist for. Perhaps there's a nice application of the FKG inequality?

Answer 1

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operatorname{\mathsf P}}$

Let $x:=(x_1,\dots,x_n)$ and $z:=(z_1,\dots,z_n)$, and assume that $\al\ge1$. It is easy to see that $x$ is independent of $\sum z_i$, so that the distribution of the random vector $x$ is the same as the conditional distribution of the random vector $z$ given $\sum z_i=1$. Also, the density of each $z_i$ is log concave. Therefore, by the following theorem by Joag-Dev and Proschan, $x$ is negatively associated (NA):

Theorem 2.8 Let $z_1,\dots,z_n$ be independent random variables with PF${}_2$ (log-concave) densities. Then the joint conditional distribution of $z_1,\dots,z_n$ given $\sum z_i$ is NA (almost surely).