Precise Deviations for the Ewens-Pitman Model

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 $α=\frac{1}{2}$.

💡 Research Summary

The paper investigates the distribution of the number of distinct types, denoted by Kₙ, in the two‑parameter Ewens‑Pitman sampling model (α∈(0,1), θ+α>0). While the law of large numbers (Kₙ/n^α → S_{α,θ} a.s.) and coarse large‑deviation and moderate‑deviation principles have been known, the authors aim to obtain precise asymptotics that retain the exact polynomial prefactors and exponential terms.

The authors first rewrite the joint distribution of the frequency counts (M₁ₙ,…,Mₙₙ,Kₙ) and observe that Kₙ can be expressed as the sum of k i.i.d. Sibuya(α) variables. Using the generating function of a Sibuya variable, they derive the compact complex‑integral representation

 P(Kₙ=k)= n! k! (θ/α)_k / (θ)n · (1/2πi)∮{|z|=r<1}