A limit theorem of discrete Galton-Watson branching processes with immigration
Zenghu Li
本文给出了离散的带移民的Galton-Watson分支过程弱收敛到连续时间和连续状态的带移民的分支过程弱收敛的一组简单的充分条件.
We provide a simple set of sufficient conditions for the weak convergence of discrete Galton-Watson branching processes with immigration to continuous time and continuous state branching processes with immigration.
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We give sufficient conditions on the offspring, the initial and the immigration distributions under which a second-order Galton-Watson process with immigration is regularly varying.
We give sufficient conditions on the initial, offspring and immigration distributions under which the distribution of a not necessarily stationary Galton--Watson process with immigration is regularly varying at any fixed time.
A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigrati…
In this short note we obtain necessary and sufficient conditions for the existence of the moments of the stationary distribution of a subcritical multitype Galton-Watson process with immigration.
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
We prove weak convergence on the Skorokhod space of Galton-Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes [Adv. Appl. Probab., 11(1979), 31--62].
Multi-type inhomogeneous Galton--Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.
Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton-Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton-Watson processes with immigration as well.
Central limit theorems for a heavy-tailed subcritical Galton-Watson process with state-dependent immigration
本文针对仅当前一代为空时发生移民的重尾次临界Galton-Watson过程，建立直至时间n总后代的中心极限定理，扩展了有限方差模型的已有结果。
Asymptotic behaviour of critical decomposable 2-type Galton-Watson processes with immigration
本文研究了后代均值矩阵可约时带移民的临界2型Galton-Watson过程的渐近行为，证明在矩条件下缩放随机阶梯过程弱收敛，极限过程由平方Bessel过程和平稳分布描述，扩展了Foster和Ney（1978）的结果。
In this paper, we consider a critical Galton-Watson branching process with immigration stopped at zero $\mathbf{W}$. Some precise estimation on the generation function of the $n$-th population are obtained, and the tail probability of the life period of $\mathbf{W}$ is studied. Based on above results, two conditional limit theorems for $\mathbf{W}$ are established.
We investigate the inhomogeneous Galton--Watson processes with immigration, where $ρ_n$ the offspring means in the $n^\textrm{th}$ generation tends to 1. We show that if the second derivatives of the offspring generating functions go to 0 rapidly enough, then the asymptotics are the same as in the INAR(1) case, treated by Györfi et al. We also determine the limit if this assumption does not hold showing the optimality of the conditions.
In this paper, we consider Galton-Watson processes with immigration. Pick $i(\ge2)$ individuals randomly without replacement from the $n$-th generation and trace their lines of descent back in time till they coalesce into $1$ individual in a certain generation, which we denote by $X_{i,1}^n$ and is called the coalescence time. Firstly, we give the probability distribution of $X_{i,1}^n$ in terms of the probability generating functions of both the offspring distribution and the immigration law. Then by studying the limit behaviors of various functionals of the Galton-Watson process with immigra…
In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(α,β)$ and of the criticality parameter $\varrho:=α+β$ for a $2$-type critical doubly symmetric positively regular Galton-Watson branching process with immigration is described.
Functional limit theorems for Galton--Watson processes with inhomogeneous immigration
本文针对具有非齐次移民的Galton--Watson过程，在子代均值极限小于或等于1时，建立了泛函极限定理，极限过程为移民极限过程的常数倍或积分泛函。
In this paper, a critical Galton-Watson branching process with immigration $Z_{n}$ is studied. We first obtain the convergence rate of the harmonic moment of $Z_{n}$. Then the large deviation of $S_{Z_n}:=\sum_{i=1}^{Z_n} X_i$ is obtained, where $\{X_i\}$ is a sequence of independent and identically distributed zero-mean random variables with tail index $α>2$. We shall see that the converging rate is determined by the immigration mean, the variance of reproducing and the tail index of $X_1^+$, comparing to previous result for supercritical case, where the rate depends on the Schröder constant …
Quasi-limiting behaviour of the sub-critical Multitype Bisexual Galton-Watson Branching Process
本文研究次临界双性Galton-Watson分支过程的拟极限行为，利用凹算子特征值与灭绝的联系，克服缺乏分支性质的困难，证明拟平稳分布的存在性与收敛性。
We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.
We consider a Galton-Watson process with immigration $(\mathcal{Z}_n)$, with offspring probabilities $(p_i)$ and immigration probabilities $(q_i)$. In the case when $p_0=0$, $p_1\neq 0$, $q_0=0$ (that is, when $\text{essinf} (\mathcal{Z}_n)$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$, of the martingale limit $\mathcal{W}$ of the process $( \mathcal{Z}_n)$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}>\text{essinf} (\mathcal{Z}_{\mathcal{K}})$ and stud…
We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $α\in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $α$-stable process if $α\in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $α= 1$.
We show the existance of the stationary distributions of subcritical multitype Galton-Watson processes without any conditions on the mean matrix of the offspring distributions. Some additional properties of the stationary distribution are also proven.
We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton--Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton--Watson trees with finitely many types.
Coalescence times for critical Galton-Watson processes with immigration
本文针对带移民的临界 Galton-Watson 过程，考察第 n 代随机两粒子及整个种群的溯祖时间在 n 趋于无穷时的极限行为。
We study certain consistent families $(F_λ)_{λ\ge 0}$ of Galton-Watson forests with lifetimes as edge lengths and/or immigrants as progenitors of the trees in $F_λ$. Specifically, consistency here refers to the property that for each $μ\leλ$, the forest $F_μ$ has the same distribution as the subforest of $F_λ$ spanned by the black leaves in a Bernoulli leaf colouring, where each leaf of $F_λ$ is coloured in black independently with probability $μ/λ$. The case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian c…
In this note we propose a two-spine decomposition of the critical Galton-Watson tree and use this decomposition to give a probabilistic proof of Yaglom's theorem.
Small deviations for critical Galton-Watson processes with infinite variance
本文研究后代方差无限的临界 Galton-Watson 过程的小偏差概率渐近性质，并以此分析约化临界 Galton-Watson 过程的结构。
We analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.
The Galton-Watson process belongs to standard teaching in probability. The basic theorem says that survival is only possible when the fecondity exceeds 1. The classical proof is essentially analytic, using generating functions and convexity arguments. We give here a more probabilistic proof, in the spirit of Bezuidenhout and Grimmett.
We prove that the speed of a $λ$-biased random walk on a supercritical Galton-Watson tree is differentiable for $λ$ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain $2$-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.
We prove that the speed of $λ$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $λ\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
We investigate conditioning Galton-Watson trees on general recursive-type events, such as the event that the tree survives until a specific level. It turns out that the conditioned tree is again a type of Galton-Watson tree, with different types of offspring and a level-dependent offspring distribution, which will all be given explicitly. As an interesting application of these results, we will calculate the expected cost of searching a tree until reaching a given level.
Population size of critical Galton-Watson processes under small deviations and infinite variance
本文研究了在观察时刻n处种群大小异常小的假设下，临界Galton-Watson过程在无限方差情况下的种群大小分布演变。
Survival Probabilities for N-ary Subtrees on a Galton-Watson Family Tree
Galton-Watson分支过程的家谱树可以包含N元子树，即其顶点具有至少N>0个子的子树。对于没有无限N元子树的家族树，我们研究了高度为t的N元子树在t趋于无穷大时消失的速度。
We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $λ$-biased random walk on Galton-Watson trees, for which the invarian…
An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees
1981年，J. Hawkes在分支测度的总质量的无穷远处的尾的一定假设下，给出了超临界Galton-Watson树边界的Hausdorff度规函数的精确形式。Hawkes猜想已被T. Watanabe在2007年以及其他关于Galton-Watson树边界分形性质的精确结果。本文的目的是在比T. Watanabe的论文，使用Lyons，Pemantle和Peres在1995年提出的有大小偏差的Galton-Watson树。
Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong ratio limit theorem for aperiodic random walks on Z^d .
We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.
We study the critical parameter u^{*} of random interlacements percolation (introduced by A.S Sznitman in arXiv:0704.2560) on a Galton-Watson tree conditioned on the non-extinction event. Starting from the previous work of A. Teixeira in arXiv:0907.0316, we show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We also characterize this value as the solution of a certain equation.
We propose a new way to condition random trees, that is, condition random trees to have large maximal out-degree. Under this new conditioning, we show that conditioned critical Galton-Watson trees converge locally to size-biased trees with a unique infinite spine. For the sub-critical case, we obtain local convergence to size-biased trees with a unique infinite node. We also study tail of the maximal out-degree of sub-critical Galton-Watson trees, which is essential for the proof of the local convergence.
Heavy-tailed critical Galton--Watson processes with immigration
针对后代分布属(1+α)-稳定律吸引域、移民分布满足两类条件的带移民临界Galton--Watson过程，证明其平稳分布尾部正则变化，确定尾部过程，建立部分和的稳定中心极限定理，且过程极值指数为0。
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Addi…
Ben Arous, Fribergh and Sidoravicius \cite{GAV2014} proved that speed of biased random walk $RW_λ$ on a Galton-Watson tree without leaves is strictly decreasing for $λ\leq \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"ıdékon \cite{EA2013} improved this result to $λ\leq \frac{1}{2}.$ In this paper, we prove that for the $RW_λ$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $λ\in \left[0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"ıdékon \cite{EA2013}.
Recently, Abraham and Delmas constructed the distributions of super-critical Lévy trees truncated at a fixed height by connecting super-critical Lévy trees to (sub)critical Lévy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.
We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.
We give an expression of the speed of the biased random walk on a Galton--Watson tree. In the particular case of the simple random walk, we recover the result of Lyons, Pemantle and Peres \cite{LyPePe95}. The proof uses a description of the invariant distribution of the environment seen from the particle.
We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.
We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.
Invariant Galton-Watson (IGW) tree measures is a one-parameter family of critical Galton-Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are the only attractors of critical Galton-Watson measures wit…
In this note we consider both the local maximal out-degree and the global maximal out-degree of Galton-Watson trees. In particular, we show that the tail of any local maximal out-degree and that of the offspring distribution are asymptotically of the same order. However for the global maximal out-degree, this is only true in the subcritical case.
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
Asymptotic behavior of some strongly critical decomposable 3-type Galton--Watson processes with immigration
研究后代均值矩阵为对角线为1的三角矩阵的临界可分解3型Galton-Watson带移民过程的渐近行为，证明在二阶或四阶矩条件下，适当缩放的过程弱收敛到由独立平方Bessel过程及其积分和迭代积分描述的极限过程。
In this paper, we study the time required for a λ-biased (λ>1) walk to visit all the vertices of a supercritical Galton-Watson tree up to generation n. Inspired by the extremal landscape approach in [Cortines, Louidor, Saglietti 2018] for the simple random walk on binary trees, we establish the scaling limit of the cover time in the biased setting.
We consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\leq p_1 <p_2 \leq 1$, there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$
In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.
We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form $P (x)s^x$ where P is a polynomial of degree p and s $\in$ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s $\rightarrow$ 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines.
The linear-fractional Galton-Watson processes is a well known case when many characteristics of a branching process can be computed explicitly. In this paper we extend the two-parameter linear-fractional family to a much richer four-parameter family of reproduction laws. The corresponding Galton-Watson processes also allow for explicit calculations, now with possibility for infinite mean, or even infinite number of offspring. We study the properties of this special family of branching processes, and show, in particular, that in some explosive cases the time to explosion can be approximated by …
Bisexual Galton-Watson processes are discrete Markov chains where reproduction events are due to mating of males and females. Owing to this interaction, the standard branching property of Galton-Watson processes is lost. We prove tightness for conveniently rescaled bisexual Galton-Watson processes, based on recent techniques developed by Bansaye, Caballero and M{é}l{é}ard. We also identify the possible limits of these rescaled processes as solutions of a stochastic system, coupling two equations through singular coefficients in Poisson terms added to square roots as coefficients of Brownian mo…
In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have $n$ leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have $n$ vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergenc…
We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.
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