Record indices and age-ordered frequencies in exchangeable Gibbs partitions

The frequencies X-1, X-2,... of an exchangeable Gibbs random partition Pi of N = {1, 2,...} (Gnedin and Pitman (2005)) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence i(1), i(2),... of least elements of the blocks of Pi. In particular, conditioning on 1 = i(1) < i(2) <..., a representation is shown to be X-j = xi(j-1) (infinity)Pi(i=j) (1-xi(i)) j=1,2,... where {xi(j) : j = 1, 2,...} is a sequence of independent Beta random variables. Sequences with such a product form are called neutral to the left. We show that the property of conditional left-neutrality in fact characterizes the Gibbs family among all exchangeable partitions, and leads to further interesting results on: (i) the conditional Mellin transform of X-k, given i(k), and ( ii) the conditional distribution of the first k normalized frequencies, given (k)Sigma(j=1) X-j and i(k); 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.