Ordered and unordered random partitions of an integer and the GEM distribution

On convergence and extensions of size-biased permutations

“…to the right-hand side of (3) (e.g. [9], p. 197 or [15], p. 178). This does not work if positive components of X tie with nonzero probability.…”

“…For X a random element of ,letN ′′ 1 , N ′′ 2 ... be random variables with values in N ∪{∞} such that for arbitrary k and distinct (15) if x has at least k positive components, and P(N ′′ k =∞|X = x) = 1 if the number of positive components is less than k.DefineaSBP ′′ by setting…”

Sampling Formulae Arising from Random Dirichlet Populations

“…Similarly, we shall supply a DTG formula, when sampling is from finite Dirichlet random partitions. We shall then show in each case that these sampling formulae give both ESF and DTG formulae when passing to the Kingman limit, thereby giving a new proof of these well-known results under the GEM model (see [18] for further results). To derive the pre-asymptotic DTG sampling formula, the joint law of the size-biased permutation of a Dirichlet partition will be needed.…”