Random bisection and evolutionary walks (Q2748448)

From the mathematical point of view, the idea of random bisection may be stated in simple terms. Consider the interval \((0,1).\) Choose \(X_{1}\) uniformly in this interval and let \(Y_{1}=1-X_{1}.\) Then, choose \(X_{2}\) uniformly in \((X_{1},1)\) and let \(Y_{2}=1-X_{2}.\) This process may be continued to obtain the sequences \(X_{i}\) and \(Y_{i}\) for \(i=3,4,\dots.\) Formulas for \(E(Y_{n}^{\nu})\) and \(E(X_{n}^{\nu})\) are known for every integer \(\nu \geq 1.\) The formula for \(E(X_{n}^{\nu})\) is difficult to evaluate numerically but the author develops an asymptotic formula for it as \(\nu \to \infty.\)NEWLINENEWLINEAmong other things, this formula is used to relate evolutionary random walk to a bisection process, familiar in combinatorics and number theory, which may be thought of as a transformed Poisson process. A through treatment of the length of the walk is given, by computing its distribution and moment generating function. It is shown that the walk length is asymptotically normally distributed. The author also treats it as a mixture of Poisson random variables and the asymptotic distribution of the Poisson parameter is derived. In concluding the paper, the author shows that jumps in the walk ordered by size have the same asymptotic distribution as normalized cycle lengths in a random permutation.NEWLINENEWLINERemark: The author mentions some possible applications to molecular evolution but does not give any concrete examples, even though some works from biology are cited.