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CALIBRATING THE CLOCK: USING STOCHASTIC PROCESSES TO MEASURE THE RATE OF EVOLUTION 145 structure to the ESF process , say. The approximating process is still discrete and, although not independent, it has a simpler structure than the original process. For random polynomials, it is shown in Arratia et al. (1993) that (5.38) so that the counts of factors of large degree can indeed be compared successfully to the corresponding counts for the ESF. The estimate in (5.38) has as a consequence the fact that the (renormalized) factors of largest degree have asymptotically the PD(1) law, a result that also follows from work of Hansen (1994). In addition, a rate of convergence is also available. In fact, (5.38) essentially holds for any of the logarithmic class (cf. Arratia et al., 1994a). In conclusion, we have seen that a variety of interesting functionals of the component structure of certain combinatorial processes can be approximated in total variation norm by either those functionals of an independent process or those functionals of the ESF itself. The important aspect of this is the focus on discrete approximating processes, rather than those found by renormalizing to obtain a continuous limit. In a very real sense, our knowledge of ''the biology of random permutations," as described by the ESF, has provided a crucial ingredient in one area of probabilistic combinatorics. WHERE TO NEXT? In the preceding sections, we have illustrated how coalescent techniques can be used to model the evolution of samples of selectively neutral DNA sequence data. Simple techniques for estimating substitution rates, some based on likelihood methods and some on more ad hoc moment methods, were reviewed. We also illustrated how the probabilistic structure of the coalescent might be used to simulate observations in order to assess the variability of such estimators.
