Record Indices and Age-Ordered Frequencies in Exchangeable Gibbs Partitions

Abstract

The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence of record indices (i.e. the least elements) of the blocks of the partition. In particular we show that, conditionally on the record indices, the distribution of the age-ordered frequencies has a left-neutral stick-breaking structure. Such a property in fact characterizes the Gibbs family among all exchangeable partitions and leads to further interesting results on: (i) the conditional Mellin transform of the $k$-th oldest frequency given the $k$-th record index, and (ii) the conditional distribution of the first $k$ normalized frequencies, given their sum and the $k$-th record index; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens' partitions.