Record indices and age-ordered frequencies in Exchangeable Gibbs Partitions

We consider a random partition Π of N = {1, 2, . . .} such that, for each n, its restriction Πn to [n] = {1, . . . , n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n− αk)Vn+1,k + Vn+1,k+1 k ≤ n = 1, 2, . . . (Gnedin and Pitman 2006). By ranking the blocks Πn1, . . . , Πnk of Πn by their age-order i.e. by the order of their least elements i1, . . . , ik, we study how the distribution of the frequencies of the blocks depends on i1, . . . , ik. Several interesting representations for the limit age-ordered relative frequencies X1, X2, . . . of Π arise, depending on which ij ’s one conditions on. In particular, conditioning on the entire vector i = 1 = i1 < i2 < . . ., a representation is Xj = ξj−1 ∞ ∏ i=j (1− ξi) j = 1, 2, . . . where the ξj ’s are independent Beta random variables with parameters, respectively, (1−α, ij+1−αj−1). We show the connection of such a representation with the so-called Beta-Stacy class of random discrete distributions (Walker and Muliere 1997). The vector i is found to form a Markov chain depending on both α and V . When V is chosen from Pitman’s subfamily, the two-parameter GEM distribution is reobtained by averaging the ξ over i . Conditioning on ik alone, we give two alternative representations for the Laplace transform of both− logXk and− log( ∑k i=1Xi), and we characterize Ewens’ partitions as the only exchangeable Gibbs partitions for which − logXk|ik can be represented as an infinite sum of independent random variables. We finally show that, for every k, conditional on ∑k i=1Xi, the distribution of the normalized age-ordered frequencies X1/ ∑k i=1Xi, . . . , Xk/ ∑k i=1Xi is a mixture of Dirichlet distributions on the (k − 1)-dimensional simplex, whose mixing measure is indexed by ik. We provide a non-trivial explicit formula for the marginal distribution of ik. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens’ partitions.