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?



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


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): 



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> [Theorem 2.8](https://www.jstor.org/stable/2240482) 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). 
> 
> 
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