Precise Deviations for the Ewens-Pitman Model

TL;DR

This paper derives an integral representation for the distribution of the number of types in the Ewens-Pitman model and establishes precise large and moderate deviations, revealing a phase transition at a critical point.

Contribution

It introduces a new integral representation for the distribution of $K_n$ and characterizes the phase transition in the rate function.

Findings

Rate function exhibits a second-order phase transition.

Critical point identified at alpha=1/2.

Precise large and moderate deviation results obtained.

Abstract

In this paper, we derive an integral representation for the distribution of the number of types $K_n$ in the Ewens-Pitman model. Based on this representation, we also establish precise large deviations and precise moderate deviations for $K_n$. After careful examination, we find that the rate function exhibits a second-order phase transition and the critical point is $\alpha=\frac{1}{2}$.