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

TL;DR

The paper addresses large deviations for the partition count $K_n$ in the Ewens–Pitman model with $\alpha\in(0,1)$ and $\theta>-\alpha$ by introducing an integral representation of the moment-generating function in terms of the Mittag–Leffler function. This approach yields a concise proof of the Feng–Hoppe large deviation principle for $K_n/n$, with the rate function $I_\alpha$ obtained via the Legendre transform of $L_\alpha(t)$, and it extends to $\theta\neq0$ cases through careful bounding arguments. Additionally, the authors establish a sharp concentration inequality involving the same rate function, offering a unified framework for both large deviations and concentration phenomena in this model. The methodology clarifies the role of the Mittag–Leffler structure in the mgf and suggests avenues for sharper deviations and extensions to related partition statistics and functionals.

Abstract

The Ewens-Pitman model is a probability distribution for random partitions of the set $[n]=\{1,\ldots,n\}$, parameterized by $α\in[0,1)$ and $θ>-α$, with $α=0$ 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 $α\in(0,1)$ and $θ>-α$. 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 $α$. 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.

Theorems & Definitions (6)

• Theorem 1.1: Feng-Hoppe large deviation principle
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof