Random walk and the existence of evolutionary rates

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Abstract
Before one can study evolutionary rates one must reject the null model of symmetric random walk. for which the requisite quantity does not exist. As random walks reliably simulate all the features we find so compelling in the fossil record—jumps, trends, and irregular cycles—rejection of this irritating hypothesis is much more difficult than one might hope. This paper reviews principal theorems from the mathematical literature of random walk and shows how they may be applied to empirical data by scaling net changes according to the square root of elapsed time. The notorious pair of “opposite” findings, equilibrium and anagenesis, may be construed as deviations from random walk in opposite directions. Malmgren's data on Globorotalia tumida, previously interpreted as an example of punctuated anagenesis, are consistent with a random walk showing neither punctuation nor anagenesis, but instead varying in speed over four subsequences.