The distribution of the number of distinct values in a finite exchangeable sequence

Abstract

Let $K_n$ denote the number of distinct values among the first n terms of an infinite exchangeable sequence of random variables $(X_1,X_2,\dots )$. We prove for $n=3$ that the extreme points of the convex set of all possible laws of $K_3$ are those derived from i.i.d. sampling from discrete uniform distributions and the limit case with $\mathbb{P}(K_3=3)=1$. We also consider the problem in higher dimensions and variants of the problem for finite exchangeable sequences and exchangeable random partitions.