By Sheldon Ross

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We consider a population composed of various numbers of different types (for example mutants, alleles, in a biological context) which is evolving continuously in time. There is an input process I(t) describing how new mutants enter the population and a stochastic structure x(t) with x(0) = 1 and the convention x(t) = 0 if t < 0, prescribing the growth pattern of each mutant population. Mutants arrive at the times 0 ≤ T1 < T2 < · · · and initiate lines according to independent versions of x(t). Thus let {xi (t)} be independent copies of x(t) with xi (t) being initiated by the ith mutant.

An (τn )) is a random partition of n. Based on our construction we have P{An+1 = (a1 + 1, . . , an , 0) | An = (a1 , . . , an )} θ = ; n+θ if ai ≥ 1, 1 ≤ i < n, P{An+1 = (. . , ai − 1, ai+1 + 1, . . , 0) | An = (a1 , . . , an )} iai = ; n+θ P{An+1 = (a1 , . . , an−1 , 0, 1) | An = (a1 , . . 31) one of the existing mutant populations is increased by 1. As with Hoppe’s urn, we have established the following theorem. 16. The distribution of the random partition An = (a1 (τn ), . . , an (τn )) is given by the Ewens sampling formula.

Ci . )ci . 53) i=1 Finally, to discount the repetition of ci cycles, the total count needs to be divided by n ∏ ci !. 54) i=1 The theorem is now proved by noting that the probability P{C1 (n) = c1 , . . 54). 8 The Dirichlet Process The Poisson–Dirichlet distribution is the distribution of allelic frequencies in descending order and individual type information is lost. Thus it is called unlabeled. Let S be a compact metric space, and ν0 a probability measure on S. The labeled version of the Poisson–Dirichlet distribution is a random measure with mean measure ν0 .