A new look on large deviations and concentration inequalities for the Ewens-Pitman model

Abstract

The Ewens-Pitman model is a probability distribution for random partitions of the set $[n]=\{1,\ldots,n\}$, parameterized by $\alpha\in[0,1)$ and $\theta>-\alpha$, with $\alpha=0$ corresponding to the Ewens model in population genetics. The goal of this paper is to provide an alternative and concise proof of the Feng-Hoppe large deviation principle for the number $K_{n}$ of partition sets in the Ewens-Pitman model with $\alpha\in(0,1)$ and $\theta>-\alpha$. Our approach leverages an integral representation of the moment-generating function of $K_{n}$ in terms of the (one-parameter) Mittag-Leffler function, along with a sharp asymptotic expansion of it. This approach significantly simplifies the original proof of Feng-Hoppe large deviation principle, as it avoids all the technical difficulties arising from a continuity argument with respect to rational and non-rational values of $\alpha$. Beyond large deviations for $K_{n}$, our approach allows to establish a sharp concentration inequality for $K_n$ involving the rate function of the large deviation principle.