60F05 Central limit and other weak theorems
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- Cannings model (1)
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We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new Cannings ancestral selection graph in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane’s formula states that for a single selectively advantageous individual in a population of haploid individuals of size N the probability of fixation is asymptotically (as N→∞) equal to the selective advantage of haploids sN divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences sN obeying N−1≪sN≪N−1/2, which is a regime of “moderately weak selection”. It turns out that for sN≪N−2/3 the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.
In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with n black balls. Empty it in n steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers Uk, 0 < k < n, of red balls after k steps exhibit an unexpected property: (U0, ... ,Un) and (Un, ... ;U0) are equal in distribution.
The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.