Petersburg Department of Steklov Institute of Mathematics

PREPRINT 3/2001

ON THE SIMPLEST SPLIT-MERGE OPERATOR ON THE INFINITE-DIMENSIONAL SIMPLEX

This preprint was accepted May 31, 2001
Contact: Natalia TSILEVICH

ABSTRACT:
We consider the simplest split-merge Markov operator $T$ on the
infinite-dimensional simplex $\Si_1$ of monotone non-negative sequences
with unit sum. For a sequence $x\in\Si_1$, 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
$(1,0,0,{\ldots}$) converge to the Poisson--Dirichlet distribution $PD(1)$.