Asympotic behavior of the total length of external branches for Beta-coalescents

We consider a ${\Lambda}$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. ${\Lambda}={\delta}_0$, the result is well known and is useful, together with the total length $L^{(n)}$, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures ${\Lambda}$, including Beta$(2-{\alpha},{\alpha})$ with $0<\alpha<1$, M{\"o}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure ${\Lambda}$ is Beta$(2-{\alpha},{\alpha})$, with $1<\alpha<2$. We prove that $n^{{\alpha}-2}L^{(n)}_{ext}$ converges in $L^2$ to $\alpha(\alpha-1)\Gamma(\alpha)$. As a consequence, we get that $L^{(n)}_{ext}/L^{(n)}$ converges in probability to $2-\alpha$. To prove the asymptotics of $L^{(n)}_{ext}$, we use a recursive construction of the $n$-coalescent by adding individuals one by one. Asymptotics of the distribution of $d$ normalized external branch lengths and a related moment result are also given

On a Markov chain related to the individual lengths in the recursive construction of Kingman's coalescent

Kingman's coalescent is a widely used process to model sample genealogies in population genetics. Recently there have been studies on the inference of quantities related to the genealogy of additional individuals given a known sample. This paper explores the recursive (or sequential) construction which is a natural way of enlarging the sample size by adding individuals one after another to the sample genealogy via individual lineages to construct the Kingman's coalescent. Although the process of successively added lineage lengths is not Markovian, we show that it contains a Markov chain which records the information of the successive largest lineage lengths and we prove a limit theorem for this Markov chain.Comment: 13 pages, 2 figure

Kingman's model with random mutation probabilities: convergence and condensation II

A generalisation of Kingman’s model of selection and mutation has been made in a previous paper which assumes all mutation probabilities to be i.i.d.. The weak convergence of fitness distributions to a globally stable equilibrium was proved. The condensation occurs if almost surely a positive proportion of the population travels to and condensates on the largest fitness value due to the dominance of selection over mutation. A criterion of condensation was given which relies on the equilibrium whose explicit expression is however unknown. This paper tackles these problems based on the discovery of a matrix representation of the random model. An explicit expression of the equilibrium is obtained and the key quantity in the condensation criterion can be estimated. Moreover we examine how the design of randomness in Kingman’s model affects the fitness level of the equilibrium by comparisons between different models. The discovered facts are conjectured to hold in other more sophisticated models

On a representation theorem for finitely exchangeable random vectors

A random vector $X=(X_1,\ldots,X_n)$ with the $X_i$ taking values in an arbitrary measurable space $(S, \mathscr{S})$ is exchangeable if its law is the same as that of $(X_{\sigma(1)}, \ldots, X_{\sigma(n)})$ for any permutation $\sigma$. We give an alternative and shorter proof of the representation result (Jaynes \cite{Jay86} and Kerns and Sz\'ekely \cite{KS06}) stating that the law of $X$ is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite $S$. The passing from finite $S$ to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof ("canonical mixing measure") by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on $S$, but for $S=\mathbb{R}$, one can choose a mixing measure on $\mathbb{R}^n$.Comment: We here give an alternative proof of the measurability of the random signed-measure underlying the construction. We also add an independent proof of the main algebraic fact used in the paper. Title update

Polynomial approximations to continuous functions and stochastic compositions

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.Comment: 21 pages, 5 figure

Does the ratio of Laplace transforms of powers of a function identify the function?

We study the following question: if $f$ is a nonzero measurable function on $[0,\infty)$ and $m$ and $n$ distinct nonnegative integers, does the ratio $\widehat{f^n}/\widehat{f^m}$ of the Laplace transforms of the powers $f^n$ and $f^m$ of $f$ uniquely determine $f$? The answer is yes if one of $m, n$ is zero, by the inverse Laplace transform. Under some assumptions on the smoothness of $f$ we show that the answer in the general case is also affirmative. The question arose from a problem in economics, specifically in auction theory where $f$ is the cumulative distribution function of a certain random variable. This is also discussed in the paper

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness

The Lenski experiment investigates the long-term evolution of bacterial populations. Its design allows the direct comparison of the reproductive fitness of an evolved strain with its founder ancestor. It was observed by Wiser et al. (2013) that the relative fitness over time increases sublinearly, a behaviour which is commonly attributed to effects like clonal interference or epistasis. In this paper we present an individual-based probabilistic model that captures essential features of the design of the Lenski experiment. We assume that each beneficial mutation increases the individual reproduction rate by a fixed amount, which corresponds to the absence of epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part of the model. Using an approximation by near-critical Galton-Watson processes, we prove that under some assumptions on the model parameters which exclude clonal interference, the relative fitness process converges, after suitable rescaling, in the large population limit to a power law function.Comment: minor changes, additional references, some comments on the notion of relative fitness and on the modelling assumptions adde

Extremal shot noise processes and random cutout sets

We study some fundamental properties, such as the transience, the recurrence, the first passage times and the zero-set of a certain type of sawtooth Markov processes, called extremal shot noise processes. The sets of zeros of the latter are Mandelbrot's random cutout sets, i.e. the sets obtained after placing Poisson random covering intervals on the positive half-line. Based on this connection, we provide a new proof of Fitzsimmons-Fristedt-Shepp Theorem which characterizes the random cutout sets