Distribution of $i$th largest entry in multinomial sample

Question

I have a $k$-class multinomial distribution with vector of probabilities $(p_1, p_2, \ldots, p_k)$, from which I draw a size-$N$ sample $(c_1, c_2, \ldots, c_k), \sum\limits_{s=1}^k c_s = N$. Assume for simplicity that $c_j \geq 1$ for all $j$. Next, I sort the sample in descending order by count to obtain $(c_{(1)}, c_{(2)}, \ldots, c_{(k)})$ where $c_{(\ell)} \geq c_{(\ell+1)}$.

Are there any research papers or other resources that describe the distribution over $c_{(i)}$, the $i$th largest count in a multinomial sample? For example, the distribution over $c_{(1)}$ is the distribution over the largest count and $c_{(k)}$ is the distribution over the smallest.

Or, asked a different way, what is the distribution describing the probability that the $j$th class's count is the $i$th largest in the sample? I've been googling around for a while but haven't really found anything relevant.

Answer 1

The distribution of $i$-th largest entry in multinomial sample is described in «Exact distributions of multinomial order statistics» by Anton Ogay. The formulations of his results, however, are too long to be reposted there.

P.S.: Yes, I know, that it is a link only answer. Its main purpose is to remove the question with known answer from the «unanswered» queue. Please, do not delete it, unless you are going to post something better.