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

TL;DR

This paper characterizes the possible distributions of the number of distinct values in small exchangeable sequences, revealing that extremal cases are derived from i.i.d. uniform sampling and a specific limit case.

Contribution

It provides a complete characterization for the case n=3 and proposes a conjecture for larger n regarding the distribution of distinct values in exchangeable sequences.

Findings

Extreme points for n=3 are from i.i.d. uniform distributions and a limit case.

Characterization of the laws of K_3 in exchangeable sequences.

Conjecture for the structure of laws for larger n.

Abstract

Let $K_n$ denote the number of distinct values among the first $n$ terms of an infinite exchangeable sequence of random variables $(X_1,X_2,\ldots)$. 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 $P(K_3=3)=1$, and offer a conjecture for larger $n$. We also consider variants of the problem for finite exchangeable sequences and exchangeable random partitions.