The number of small blocks in exchangeable random partitions
نویسنده
چکیده
Suppose Π is an exchangeable random partition of the positive integers and Πn is its restriction to {1, . . . , n}. Let Kn denote the number of blocks of Πn, and let Kn,r denote the number of blocks of Πn containing r integers. We show that if 0 < α < 1 and Kn/(n l(n)) converges in probability to Γ(1 − α), where l is a slowly varying function, then Kn,r/(n l(n)) converges in probability to αΓ(r − α)/r!. This result was previously known when the convergence of Kn/(n l(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 Kn grows only slightly slower than n fails to be true.
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