We demonstrate a novel strong law of large numbers for branching processes, with a simple proof via measure-theoretic manipulations and spine theory. Roughly speaking, any sequence of events that eventually occurs almost surely for the spine entails the almost sure convergence of a certain sum over particles in the population.
In the literature, the spine decomposition of branching Markov processes was constructed under the assumption that each individual has at least one child. In this paper, we give a detailed construction of the spine decomposition of general branching Markov processes allowing the possibility of no offspring when a particle dies. Then we give some applications of the spine decomposition.
Spine proofs for Lp-convergence of branching-diffusion martingales
利用Hardy和Harris（2006）[“A new formulation of the spine approach in branching diffusions”，arXiv：math.PR/0611054]中奠定的基础，我们对三种不同的分支扩散模型给出了一些关键“加法”鞅的L^p-收敛p>=1）的新的spine证明，包括多类型分支布朗运动的新结果和最左粒子速度的讨论。我们开发的脊柱技术在Kyprianou（2004）[“KPP方程的行波解，Ann.Inst.H.2004]中发现的概念脊柱证明的精神中给出了清晰和简单的论点。庞加莱概率中央集权主义者40，no.1，pp53-72]和Lyons等人（1997）[“A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes”，Classical and modern branching processes，IMA Vol. Math. Appl.，vol.84，Springer，New York，pp 181 -185]，并且它们也应该扩展到更一般的分支扩散类。重要的是，本文中的技术还为Hardy和Harris（2006）[“A conceptual approach to a path result for branch…
In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements the related results obtained in Evans (1993), Kyprianou et al. (2012) and Liu, Ren and Song (2009) for superprocesses with purely local branching mechanisms and in Chen, Ren and Song (2016) and Kyprianou and Palau (2016) for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particul…
We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts, assuming the tails of the displacement decays at exponential rate; Bérard and Maillard took interest in the case of heavy tail displacements. We take interest in an …
A new formulation of the spine approach to branching diffusions
我们提出了分支扩散的尺度方法的脊变化的形式化，其改进了Kyprianou（2004）[“KPP方程的行波解，Ann.Inst.H.庞加莱概率中央集权主义者40，no.1，pp 53 -72]，其本身利用了Lyons等人（1997）的早期作品[“A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes”，Classical and modern branching processes，IMA Vol. Math. Appl.，第84卷，Springer，纽约，第181 -185页]。我们使用我们的新配方来解释某些“吉布斯-玻尔兹曼”粒子的权重，并使用它来给出一个新的，直观的和证明一个更一般的“多对一”的结果，使期望的总和在分支扩散的粒子计算纯粹在一个粒子的期望。值得注意的是，我们的形式化提供了基础，促进了分支扩散中各种新的、大大简化的和更直观的证明：例如，参见Hardy和Harris（2006）中的加法鞅的L^p收敛[“Spine proofs for L^p-convergence of branching-diffusion martingales”，arXiv：math.PR/0611056],分支布朗运动的路径大偏差结果在Hardy和Harris（2006）[”A co…
We consider branching random walks with a spine in the domain of attraction of an $α$-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance …
We prove local convergence results of rerooted conditioned multi-type Galton--Watson trees. The limit objects are multitype variants of the random sin-tree constructed by Aldous (1991), and differ according to which types recur infinitely often along the backwards growing spine.
Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration (CBI process) converges almost surely. If an $x \log(x)$ moment condition on the branching mechanism does not hold, then the limit is zero. If this $x \log(x)$ moment condition holds, then we prove $L_1$ convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing. If, in addition, a suitable extra power moment condi…
In Li (2011), Example 2.2, the notion of a multi-type continuous-state branching process (MCSBP) was introduced with a finite number of types, with the countably infinite case being proposed in Kyprianou and Palau (2017). One may consider such processes as a super-Markov chain on a countable state-space of types, which undertakes both local and non-local branching. In Kyprianou and Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists a lead eigenvalue which characterises the spectral radius of the linear semigroup associated to the process. Moreover, in a qualitative…
Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E}\log m(ξ_{0})=\infty$. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left(\barξ,Z_{n}(\barξ)\right)$ converges almost surely to a random variable $Y(\barξ)$, where $Y\in(0,1)~η$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\left\{\frac{U(\barξ,Z_{n}(\bar…
From multitype branching Brownian motions to branching Markov additive processes
University of Chinese Academy of Sciences（中国科学院大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University（北京大学数学学院（LMAM）及统计科学中心）；School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学科学学院）
We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that if and only if the branching rate is sufficiently large, then the population survives forever with positive probability. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean population size converges to an explicit limit, almost surely and in $L^1$. As a consequence, we get that al…
Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to infinity.
Consistent Minimal Displacement of Branching Random Walks
令$\mathbb{T}$表示有根$b$叉树，并且令$\{S_v\}_{v\in \mathbb{T}}$表示由树的顶点索引的分支随机游走，其中增量是i.i.d.。并具有对数矩母函数$Λ(\cdot)$。令$m_n$表示在$n$第代的所有顶点上的变量$S_v$的最小值，由$\mathbb{D}_n$表示。在温和的条件下，$m_n/n$几乎必然收敛于一个常数，为了方便起见，可以取为0。使用$\bar S_v=\max\{S_w:{\rm $ w $ is on the geodesic connecting the root to $ v $}\}$，定义$L_n=\min_{v\in \mathbb{D}_n} \bar S_v$。证明了$L_n/n^{1/3}$几乎必然收敛于一个显式常数$l_0$ .这回答了胡和史的一个问题。
Let $η^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $η_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $η^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest.
We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is $z\geq 1$. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation $n$ and let $n\to\infty$; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the clas…
We consider the branching random walk in random environment with a random absorption wall. When we add this barrier, we discuss some topics related to the survival probability. We assume that the random environment is i.i.d., $S_i$ is a particular i.i.d. random walk depend on the random environment $\mathcal{L}$. Let the random barrier function (the random absorption wall) is $g_i(\mathcal{L}):=ai^α-S_i,$ where $i$ present the generation. We show that there exists a critical value $a_c>0$ such that if $a>a_c,α=\frac{1}{3}$, the survival probability is positive almost surly and if $a<a_c,α=\fra…
A continuous-state branching process in varying environments is constructed by the pathwise unique solution to a stochastic integral equation driven by time-space noises. The process arises naturally in the limit theorem of Galton--Watson processes in varying environments established by Bansaye and Simatos (2015). In terms of the stochastic equation we clarify the behavior of the continuous-state process at its bottlenecks, which are the times when it arrives at zero almost surely by negative jumps.
The neutron transport equation (NTE) describes the flux of neutrons across a planar cross-section in an inhomogeneous fissile medium when the process of nuclear fission is active. Classical work on the NTE emerges from the applied mathematics literature in the 1950s through the work of R. Dautray and collaborators, [7, 8, 19]. The NTE also has a probabilistic representation through the semigroup of the underlying physical process when envisaged as a stochastic process; cf. [7, 17, 18, 20]. More recently, [6] and [16] have continued the probabilistic analysis of the NTE, introducing more recent…
In this work we study the bisexual Galton-Watson process with a finite number of types, where females and males mate according to a ''mating function'' and form couples of different types. We assume that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. In the absence of a linear reproduction operator which is the key to understand the behaviour of the model in the asexual case, we build a concave reproduction operator and use a concave Perron-Frobenius theory to ensure the exis…
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineag…
Let $G$ be a Cayley graph of a nonamenable group with spectral radius $ρ< 1$. It is known that branching random walk on $G$ with offspring distribution $μ$ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $\bar μ$ satisfies $\bar μ\leq ρ^{-1}$. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase $1<\barμ \leq ρ^{-1}$, and in particular at the recurrence threshold $\bar μ= ρ^{-1}$, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost sur…
The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $λz$ for …
