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.

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