Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)

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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 144 The Large Components Thus far we have described how we might approximate a complicated dependent process (the counts of small components) by a simpler, independent process, with an estimate of the error involved. It is natural to ask what can be said about the large component counts. To describe this, we return once more to the ESF. Let L1 ≡ L1 (n) ≥ L2 ≥···≥ LK denote the sizes of the largest cycle, the second largest cycle, . . . , the smallest of the random number of cycles in a q-biased random permutation. We define Lj = Lj(n) = 0, j > K. It is known from the work of Kingman (1974, 1977) that the random vector n−1 (L1,L2,. . .,LK,0,0,. . .) converges in distribution to a random vector (X1, X2,. . .). The vector X has the Poisson-Dirichlet distribution with parameter θ, which we denote by PD(θ). There are a number of characterizations of PD(θ), among them Kingman's original definition: Let σ1 ≥ σ2 ≥ ···≥ 0 denote the points of a Poisson process on (0,∞) having mean measure with density 0 e−x / x, x > 0, and set σ = ∑i≥1 σi . Then We know that the large components, those that are of a size about n, of a θ-biased random permutation are described asymptotically by the PD(θ) law. What can be said about the large components of the other combinatorial structures we have seen? We focus once more on the logarithmic structures that satisfy either condition (5.31) or (5.37), where population genetics has a crucial role to play once more. In approximating the behavior of counts of large components Cr = (Cr+l, Cr+2,. . .,Cn) we should not expect to be able to compare to an independent process because, for example, there can be at most components of size j or greater, and this condition forces very strong correlations on the counts of large components. However, we should be able to compare the component counting process Cr of the combinatorial