On the simplest split-merge operator on the infinite-dimensional simplex

TL;DR

This paper analyzes a simple split-merge Markov operator on the infinite-dimensional simplex, proving that its trajectory starting from a specific measure converges to the Poisson-Dirichlet distribution PD(1).

Contribution

It introduces and studies the simplest split-merge operator on the infinite-dimensional simplex, establishing convergence to PD(1) from a specific initial measure.

Findings

Trajectory converges to Poisson-Dirichlet distribution PD(1)

Operator involves size-biased sampling, merging, and splitting

Convergence proven for the delta measure at (1,0,0,...)

Abstract

We consider the simplest split-merge Markov operator $T$ on the infinite-dimensional simplex $\Sigma_1$ of monotone non-negative sequences with unit sum. For a sequence $x\in\Sigma_1$, it picks a size-biased sample (with replacement) of two elements of $x$; if these elements are distinct, it merges them and reorders the sequence, and if the same element is picked twice, it splits this element uniformly into two parts and reorders the sequence. We prove that the means along the $T$-trajectory of the $\de$-measure at the vector $(1,0,0,{...})$ converge to the Poisson--Dirichlet distribution PD(1).