Supercritical Galton-Watson Process Research Articles

Polygenic dynamics underlying the response of quantitative traits to directional selection

We study the response of a quantitative trait to exponential directional selection in a finite haploid population, both at the genetic and the phenotypic level. We assume an infinite sites model, in which the number of new mutations per generation in the population follows a Poisson distribution (with mean Θ) and each mutation occurs at a new, previously monomorphic site. Mutation effects are beneficial and drawn from a distribution. Sites are unlinked and contribute additively to the trait. Assuming that selection is stronger than random genetic drift, we model the initial phase of the dynamics by a supercritical Galton–Watson process. This enables us to obtain time-dependent results. We show that the copy-number distribution of the mutant in generation n, conditioned on non-extinction until n, is described accurately by the deterministic increase from an initial distribution with mean 1. This distribution is related to the absolutely continuous part W+ of the random variable, typically denoted W, that characterizes the stochasticity accumulating during the mutant’s sweep. A suitable transformation yields the approximate dynamics of the mutant frequency distribution in a Wright–Fisher population of size N. Our expression provides a very accurate approximation except when mutant frequencies are close to 1. On this basis, we derive explicitly the (approximate) time dependence of the expected mean and variance of the trait and of the expected number of segregating sites. Unexpectedly, we obtain highly accurate approximations for all times, even for the quasi-stationary phase when the expected per-generation response and the trait variance have equilibrated. The latter refine classical results. In addition, we find that Θ is the main determinant of the pattern of adaptation at the genetic level, i.e., whether the initial allele-frequency dynamics are best described by sweep-like patterns at few loci or small allele-frequency shifts at many. The number of segregating sites is an appropriate indicator for these patterns. The selection strength determines primarily the rate of adaptation. The accuracy of our results is tested by comprehensive simulations in a Wright–Fisher framework. We argue that our results apply to more complex forms of directional selection.

Cramér Moderate Deviations for a Supercritical Galton–Watson Process with Immigration

Cramér Moderate Deviations for a Supercritical Galton–Watson Process with Immigration

A universal right tail upper bound for supercritical Galton–Watson processes with bounded offspring

We consider a supercritical Galton–Watson process Zn whose offspring distribution has mean m>1 and is bounded by some d∈{2,3,…}. As is well-known, the associated martingale Wn=Zn/mn converges a.s. to some nonnegative random variable W∞. We provide a universal upper bound for the right tail of W∞ and Wn, which is uniform in n and in all offspring distributions with given m and d, namely: P(Wn≥x)≤c1exp−c2m−1mxd,∀n∈N∪{+∞},∀x≥0,for some explicit constants c1,c2>0. For a given offspring distribution, our upper bound decays exponentially as x→∞, which is actually suboptimal, but our bound is universal: it provides a single effective expression – which does not require large x – and is valid simultaneously for all supercritical bounded offspring distributions.

带有移民的上临界Galton-Watson过程的Cramer中偏差

带有移民的上临界Galton-Watson过程的Cramer中偏差

Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process

AbstractLet $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

Limit theorems for discrete multitype branching processes counted with a characteristic

For a discrete time multitype supercritical Galton–Watson process (Zn)n∈N and corresponding genealogical tree T, we associate a new discrete time process (ZnΦ)n∈N such that, for each n∈N, the contribution of each individual u∈T to ZnΦ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, ZnΦ is obtained by summing over all u∈T the corresponding contributions Φu, where (Φu)u∈T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristicΦ. We derive a LLN and a CLT for the process (ZnΦ)n∈N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966).

Cramér moderate deviations for a supercritical Galton–Watson process

Let (Zn)n≥0 be a supercritical Galton–Watson process. The Lotka–Nagaev estimator Zn+1/Zn is a common estimator for the offspring mean. In this paper, we establish some Cramér moderate deviation results for the Lotka–Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.

Parameterised branching processes: A functional version of Kesten & Stigum theorem

Let (Zn,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ>1 and is such that ∫xlog+(x)dμ(x)<+∞. According to the famous Kesten & Stigum theorem, (Zn/λn) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation.In this paper, we consider a family of Galton–Watson processes (Zn(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))λ∈I is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ>1 for all λ∈I. This allows us to define Zn(λ) the number of elements in the nth generation at time λ.Set Wn(λ)=Zn(λ)/λn for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (Wn(λ),λ∈I)n≥0 converges in probability as n tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.

Haldane\u2019s formula in Cannings models: the case of moderately strong selection

For a class of Cannings models we prove Haldane’s formula, pi (s_N) sim frac{2s_N}{rho ^2}, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for s_N sim N^{-b} and 0< b<1/2. Here, s_N is the selective advantage of an individual carrying the beneficial type, and rho ^2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

Deviations for Martingale Convergence of a Branching Process with Random Index

The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton–Watson process with a Poisson process as its time index. Harmonic moments and Shroder index play an important role in the proofs.

Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Consider a continuous time process {Yt=ZNt, t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1&gt;δ are estimated for three types of positive δ.

On large deviation rates for sums associated with Galton‒Watson processes

Abstract Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1&lt;∞ and X1 is in the domain of attraction of an α-stable law with 0&lt;α&lt;2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

Consider branching random walks on the integer lattice Zd, where the branching mechanism is governed by a supercritical Galton–Watson process and the particles perform a symmetric nearest-neighbor random walk whose increments equal to zero with probability r∈[0,1). We derive exact convergence rate in the local limit theorem for distributions of particles. When r=0, our results correct and improve the existing results on the convergence speed conjectured by Révész (1994) and proved by Chen (2001). As a byproduct, we obtain exact convergence rate in the local limit theorem for some symmetric nearest-neighbor random walks, which is of independent interest.

On qualitative robustness of the Lotka–Nagaev estimator for the offspring mean of a supercritical Galton–Watson process

We characterize the sets of offspring laws on which the Lotka–Nagaev estimator for the mean of a supercritical Galton–Watson process is qualitatively robust. These are exactly the locally uniformly integrating sets of offspring laws, which may be quite large. If the corresponding global property is assumed instead, we obtain uniform robustness as well. We illustrate both results with a number of concrete examples. As a by-product of the proof we obtain that the Lotka–Nagaev estimator is [locally] uniformly weakly consistent on the respective sets of offspring laws, conditionally on non-extinction.

A branching-selection process related to censored Galton–Walton processes

Nous étudions un cas particulier de système de particules avec branchement et sélection introduit par Bérard et Gouéré [Comm. Math. Phys. 298 (2010) 323–342]. Nous obtenons l’asympotique pour la vitesse, en remarquant un lien avec un processus de Galton–Watson surcritique censuré à un certain niveau.

Стохастическая модель процесса идентификации сервисов информационной системы

This article refers to labor costs estimation of information system functional requirements design problem in a service-oriented architecture. Stochastic model of information system services identification process is offered for this purpose, which allows providing the estimation of expected project objects and interrelations amount using minimal initial data. This model is based on automated business-process decomposition representation as a branching random supercritical Galton-Watson process. Decomposition elements relation design is been modeled as connected random graph construction process by Erdosh-Renyi model. Model predictions have been confirmed through verification by experimental data.

Weighted moments for the limit of a normalized supercritical Galton–Watson process

Let (Zn) be a supercritical Galton–Watson process, and let W be the limit of the normalized population size Zn/mn, where m=EZ1>1 is the mean of the offspring distribution. Let ℓ be a positive function slowly varying at ∞. Bingham and Doney (1974) [4] showed that for α>1 not an integer, EWαℓ(W)<∞ if and only if EZ1αℓ(Z1)<∞; Alsmeyer and Rösler (2004) [2] proved the equivalence for α>1 not a dyadic power. Here we prove it for all α>1.

Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

As is well known, for a supercritical Galton-Watson process Z n whose offspring distribution has mean m > 1, the ratio W n := Z n /m n has almost surely a limit, say W. We study the tail behaviour of the distributions of W n and W in the case where Z 1 has a heavy-tailed distribution, that is, \(\mathbb{E}e^{\lambda {\rm Z}_1 } = \infty \) for every λ > 0. We show how different types of distributions of Z 1 lead to different asymptotic behaviour of the tail of W n and W. We describe the most likely way in which large values of the process occur.

Time to Extinction of Galton–Watson Processes Censored at High Level

The expected hitting time into absorbing states is investigated for Markov chains generated by the supercritical Galton–Watson processes, in each generation of which only that amount of particles is preserved which does not exceed a fixed large number. A relationship is established between the components of such processes and subcritical branching processes. For subcritical branching processes initiated by a large number of particles, asymptotic representations are found for all moments of the extinction time.

On the left tail asymptotics for the limit law of supercritical Galton–Watson processes in the Böttcher case

Par un changement d’échelle bien connu, on obtient que les processus de Galton–Watson supercritiques sur Z convergent vers une variable aléatoire non-degénerée W. Nous considérons les estimées asymptotiques à gauche (près de l’origine) de la distribution. Dans le cas Böttcher (quand il y a au moins deux progénitures en chaque point), nous obtenons l’asymptotique exacte présentant un comportement oscillatoire (Théorème 1). Sous une autre hypothèse raisonnable, les oscillations s’annulent (Corollaire 2). Pour le cas Böttcher, nous présentons un résultat sur la probabilité des grandes déviations, amélioré en exprimant l’asymptotique exacte sous un scaling logarithmique (Théorème 7). En imposant d’autres conditions, nous obtenons des asymptotiques plus raffinées (Théorème 8), c’est-à-dire sans log-scaling.