A.11 Residual allocation models

A population of agents of at most countably many types is called a residual allocation model, after the model introduced by Halmos (1944), if the fractions of all types can be expressed in the form

with a sequence of independent random variables {P_i} defined in the interval [0, 1].

Engen (1975) shows that when the P's have a Dirichlet distribution D(∈, K), the distribution of the fractions tends to that of the residual allocation model in distriution, as ∈ approaches zero and K goes to infinity in such a way that Ke approaches a constant θ, when the distribution of P has the density θ (1 - p)^{θ -1}. This is called the GEM distribution by Ewens (1990, Sec. 13).

To establish this, we use a theorem of Wilks (1962, Sec. 5) to the effect that for a bounded random variable all the moments determine the distribution uniquely. We thus compare the limiting forms of the moments of the fractions and of the residual allocation models. Accordingly, suppose that P_i, i = 1, 2,..., K,is distributed according to

Let X be the fraction of the type it represents. Then, noting that the probability that X = P_i (that is, the probability that the sample is of type i is P_i), we calculate its rth moment as

In the limit as e approaches zero and K e → θ,

We thus conclude that the sequences of fractions given by the Dirichlet dis­tribution specified above tend in distribution to that specified by the residual allocation model. The converse has been established by McClosky (1965).