arXiv subjects

Linglong Yuan

Publications and source records attributed to Linglong Yuan.

27 records · Page 2Linked to original sources

arXiv2016

On the extendibility of finitely exchangeable probability measures

A length-$n$ random sequence $X_1,\ldots,X_n$ in a space $S$ is finitely exchangeable if its distribution is invariant under all $n!$ permutations of coordinates. Given $N > n$, we study the extendibility problem: when is it the case that there is a length-$N$ exchangeable random sequence $Y_1,\ldots, Y_N$ so that $(Y_1,\ldots,Y_n)$ has the same distribution as $(X_1,\ldots,X_n)$? In this paper, we give a necessary and sufficient condition so that, for given $n$ and $N$, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments that take into account the symmetry. We also address the problem of infinite extendibility. Our results are valid when $X_1$ has a regular distribution in a locally compact Hausdorff space $S$. We also revisit the problem of representation of the distribution of a finitely exchangeable sequence.

math.PR↗

arXiv2016

A generalization of Kingman's model of selection and mutation and the Lenski experiment

Kingman's model of selection and mutation studies the limit type value distribution in an asexual population of discrete generations and infinite size undergoing selection and mutation. This paper generalizes the model to analyse the long-term evolution of Escherichia. coli in Lenski experiment. Weak assumptions for fitness functions are proposed and the mutation mechanism is the same as in Kingman's model. General macroscopic epistasis are designable through fitness functions. Convergence to the unique limit type distribution is obtained.

math.PR↗

arXiv2016

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 $(X_{σ(1)}, \ldots, X_{σ(n)})$ for any permutation $σ$. We give an alternative and shorter proof of the representation result (Jaynes \cite{Jay86} and Kerns and Székely \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 $\mathbb{R}^n$.

math.PR↗

arXiv2016

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.

math.PR↗

arXiv2016

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.

math.PR↗

arXiv2014

Asymptotics of the minimal clade size and related functionals of certain beta-coalescents

This article shows the asymptotics of distributions of various functionals of the Beta$(2-α,α)$ $n$-coalescent process with $1<α<2$ when $n$ goes to infinity. This process is a Markov process taking {values} in the set of partitions of $\{1, \dots, n\}$, evolving from the intial value $\{1\},\cdots, \{n\}$ by merging (coalescing) blocks together into one and finally reaching the absorbing state $\{1, \dots, n\}$. The minimal clade of $1$ is the block which contains $1$ at the time of coalescence of the singleton $\{1\}$. The limit size of the minimal clade of $1$ is provided. To this, we express it as a function of the coalescence time of $\{1\}$ and sizes of blocks at that time. Another quantity concerning the size of the largest block (at deterministic small time and at the coalescence time of $\{1\}$) is also studied.

math.PR↗

arXiv2013

On the measure division construction of Λ-coalescents

This paper provides a new construction of Λ-coalescents called "measure division construction". This construction is pathwise and consists of dividing the characteristic measure Λinto several parts and adding them one by one to have a whole process. Using this construction, a "universal" normalization factor μ^{(n)} for the randomly chosen external branch length T^{(n)} has been discovered for a class of coalescents. This class of coalescents covers processes similar to Bolthausen-Sznitman coalescent, the coalescents without proper frequencies, and also others.

math.PR↗

arXiv2013

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

We consider a $Λ$-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. $Λ=δ_0$, 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 $Λ$, including Beta$(2-α,α)$ with $0<α<1$, M{ö}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure $Λ$ is Beta$(2-α,α)$, with $1<α<2$. We prove that $n^{α-2}L^{(n)}_{ext}$ converges in $L^2$ to $α(α-1)Γ(α)$. As a consequence, we get that $L^{(n)}_{ext}/L^{(n)}$ converges in probability to $2-α$. 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.

math.PR↗

arXiv2012

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta$(2-α,α)$ (with $1<α<2$) and related $Λ$-coalescents. If $T^{(n)}$ denotes the length of an external branch of the $n$-coalescent, we prove the convergence of $n^{α-1}T^{(n)}$ when $n$ tends to $ \infty $, and give the limit. To this aim, we give asymptotics for the number $σ^{(n)}$ of collisions which occur in the $n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the $n$-coalescent.

math.PR↗