The number of small blocks in exchangeable random partitions

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.