The number of small blocks in exchangeable random partitions

Abstract

Suppose $\Pi$ is an exchangeable random partition of the positive integers and $\Pi_n$ is its restriction to $\{1, ..., n\}$. Let $K_n$ denote the number of blocks of $\Pi_n$, and let $K_{n,r}$ denote the number of blocks of $\Pi_n$ containing $r$ integers. We show that if $0 < \alpha < 1$ and $K_n/(n^{\alpha} \ell(n))$ converges in probability to $\Gamma(1-\alpha)$, where $\ell$ is a slowly varying function, then $K_{n,r}/(n^{\alpha} \ell(n))$ converges in probability to $\alpha \Gamma(r - \alpha)/r!$. This result was previously known when the convergence of $K_n/(n^{\alpha} \ell(n))$ holds almost surely, but the result under the hypothesis of convergence in probability has significant implications for coalescent theory. We also show that a related conjecture for the case when $K_n$ grows only slightly slower than $n$ fails to be true.