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?