Lina Ji, Zenghu Li
For a positive continuous function f satisfying some standard conditions, we study the f-moments of continuous-state branching processes with or without immigration. The main results give criteria for the existence of the f-moments. The characterization of the processes in terms of stochastic equations given by Dawson and Li (2012) plays an essential role in the proofs.
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A limit theorem of discrete Galton-Watson branching processes with immigration
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.
In this paper we consider a triangular array of branching processes with non-stationary immigration. We prove a weak convergence of properly normalized branching processes with immigration to deterministic function under assumption that immigration is rowwise $ψ-$mixing and the offspring mean tends to its critical value 1, immigration mean and variance controlled by regularly varying functions. Moreover, we obtain a fluctuation limit theorem for branching process with immigration when immigration is $m-$dependent where $m$ may tend to infinity with the row index at a certain rate. In this case…
The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson ran- dom measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.
Weak convergence of continuous-state branching processes with large immigration
本文研究带大尾部移民的连续状态分支过程的弱收敛，在繁殖律一阶矩有限条件下建立泛函极限定理，并识别出四类极限过程。
Backbone decomposition for continuous-state branching processes with immigration
在Duqesne and Winkel（2007）和Berestycki et al.（2011）的精神下，我们证明了具有一般分支机制和一般迁移机制的超临界连续态分支过程在定律上等价于具有Poissonian dressing迁移的连续时间Galton Watson过程。该结果还表征了极限骨架分解，这可以从Cao和Winkel（2010）关于移民的Galton-Watson树的一致增长的工作中预测。
Branching Processes with Immigration in a Random Environment -- the Grincevičius--Grey setup
本文在移民分布主导后代分布的条件下，确定了随机环境中带移民分支过程平稳分布的尾部渐近性，其假设与随机递归方程的Grincevičius--Grey定理相同。
Stable continuous branching processes with immigration and Beta-Fleming-Viot processes with immigration
分枝过程和Fleming-Viot过程是随机种群理论中的两个主要模型。在这两个模型中引入移民现象，推广了Shiga（1990）和Birkner等人（2005）分别将Feller扩散与经典Fleming-Viot过程和α稳定连续态分支过程与β（2-alpha，alpha）-广义Fleming-Viot过程联系起来的结果.在最近的工作中，一类新的概率测度值过程，称为M-广义Fleming-Viot过程的移民，已经建立了对偶与所谓的M-聚结。本文的目的是研究这类新的过程和具有移民的连续状态分支过程之间的联系。在条件为永远不会灭绝的$α$稳定分支过程的特定情况下，我们得到它的谱系由Beta（2-alpha，alpha-1）-合并给出，直到随机时变。
A Fluctuation Limit Theorem of Branching Processes with Immigration and Statistical Applications
证明了带移民的Galton-Watson分支过程的一个一般涨落极限定理。极限是一个由谱正Levy过程驱动的非时齐的非时齐型过程。作为这个结果的应用，我们得到了后代分布和移民分布的后代均值和方差的条件最小二乘估计的一些渐近估计。
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
Branching processes with immigration and related topics
本文综述了带移民的分枝过程、广义Ornstein-Uhlenbeck过程和仿射Markov过程的研究进展。本文主要讨论了斜卷积半群的应用及其在这些过程中的联系。
We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is regularly varying generating functions with remainder.
Changing the branching mechanism of a continuous state branching process using immigration
我们构造了一个带移民的连续状态分支过程（CBI），它的移民依赖于CBI本身，我们恢复了一个连续状态分支过程（CB）.这提供了一个双重建设的剪枝在CB节点介绍了作者在以前的文件。这种构造是模拟中性突变的自然方式。利用指数公式，在总种群不灭绝的条件下，计算了原型种群在临界或次临界二次分枝中灭绝的概率.
Diffusion approximation of a sequence of critical branching processes with dependent immigration
研究具有相关迁入的临界分支过程序列的扩散近似，在一般假设下，证明其缩放和归一化序列依分布收敛到扩散过程，扩展了魏和温尼基（1989）的泛函极限定理。
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.
Asymptotic behavior of critical primitive multi-type branching processes with immigration
在自然假设下，当后代均值矩阵是本原（即正正则）时，得到了带移民的临界多型分枝过程的Feller型扩散逼近.即证明了由临界本原多型分枝过程序列构成的适当尺度的随机阶梯函数序列弱收敛于由后代均值矩阵的Perron向量确定的射线所支持的平方Bessel过程.
Multitype branching processes with immigration in random environment and polling systems
对于在随机环境中演化并产生最终产物的带移民的多型分支过程，我们发现了在系统中积累的最终产物的大小的尾分布。利用这个结果，我们研究了具有随机服务规则和随机正切换时间的分支型轮询系统的忙期的尾部分布
Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration
证明了一类带跳的随机微分方程的多型连续状态连续时间分支过程的路径唯一强解.
Parameter Estimation in Two-type Continuous-state Branching Processes with Immigration
研究了两类带移民的连续状态分支过程（CBI过程）的估计。证明了过程的遍历性。建立了基于低频观测的漂移和扩散系数的条件最小二乘估计和加权条件最小二乘估计的强相合性和中心极限定理。
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
Recurrence and transience of critical branching processes in random environment with immigration and an application to excited random walks
建立了具有移民的随机环境中临界分枝过程的常返性和瞬时性准则。这些结果然后被应用到讨论经常性的随机游动在随机环境中的Z，这将是由饼干引起的漂移的强度1的权利干扰的经常性和瞬态。
We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada--Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By spe…
Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we…
In this work we investigate limit theorems for the time-averaged process $\left(\frac{1}{t}\int_0^t X_s^x ds\right)_{t\geq 0}$ where $X^x$ is a subcritical continuous-state branching processes with immigration (CBI processes) starting in $x \geq 0$. Under a second moment condition on the branching and immigration measures we first prove the law of large numbers in $L^2$ and afterward establish the central limit theorem. Assuming additionally that the big jumps of the branching and immigration measures have finite exponential moments of some order, we prove in our main result the large deviatio…
Asymptotic behavior of critical irreducible multi-type continuous state and continuous time branching processes with immigration
在自然假设下，对具有移民的临界、不可约多型连续状态和连续时间分支过程，给出了一个Feller型扩散逼近。即证明了由临界的、不可约的多型连续态和连续时间的分支过程构成的一系列适当尺度的随机阶梯函数弱收敛于由与所讨论的分支过程的分支机制有关的矩阵的Perron向量所确定的射线所支持的平方Bessel过程。
Small value probabilities for supercritical branching processes with immigration
考虑一类带移民的超临界Galton-Watson分支过程。众所周知，在适当的条件下，后代和移民分布，有一个有限的，严格的正极限${\mathcal{W}}$的归一化总体规模。得到了${\mathcal{W}}$的小值概率。后代和移民分布之间的平衡的精确效果的特点。
Let (Zn) be a branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on logZn by using the convergence rates of the logarithm of submartingale and the result of the corresponding random walk on the Berry-Esseen bound.
A critical branching process with immigration which evolve in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the aboriginal population we investigate the tail distribution of the so-called life period of the process, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.
In this paper, we consider $n$-type Markov branching processes with immigration and resurrection. The uniqueness criteria are first established. Then, a new method is found and the explicit expression of extinction probability is successfully obtained in the absorption case, the mean extinction time is also given. The recurrence and ergodicity criteria are given if the state ${\bf 0}$ is not absorptive. Finally, if the resurrection rates are same as the immigration rates, the branching property and decay property are discussed in detail, it is shown that the process is a superimposition of a $…
Moment formulas for multi-type continuous state and continuous time branching processes with immigration
导出了多类型连续状态和连续时间带移民的分支过程的矩的递推公式。证明了$k$阶（混合）矩和$k$阶（混合）中心矩是过程初值的多项式，它们的次数至多分别为$k$和$\lfloor k/2 \rfloor$.
In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.
In this article we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d \geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a suitable comparison with one-dimensional CBI processes and an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for transience of multi-type CBI processes.
The sector constants of continuous state branching processes with immigration
研究了带移民的连续状态分支过程。我们特别关注相关的（非对称的）狄利克雷形式。在观察到这类模型的Γ分布仅是可逆分布之后，我们证明了每一个广义伽玛卷积都是具有适当选择的分支机制和连续移民的过程的平稳分布。对于这样的不可逆过程，强扇区条件进行了讨论的一个特征称为托林措施。此外，通过涉及斯蒂尔切斯变换的计算，将指出与非对易概率论概念的某些联系。
On the Limit Distributions of Continuous-State Branching Processes with Immigration
我们考虑了Kawazu和Watanabe（1971）提出的一类带移民的连续状态分支过程（CBI过程）及其在时间趋于无穷大时的极限分布。我们确定了极限分布的Levy-Khintchine三元组，并给出了Levy从属子和谱正Levy过程的特征三元组的明确描述，这些特征三元组分别描述了移民和非移民. CBI过程的分支机制。这种表示使我们能够描述极限分布的支撑，并证明其在支撑边界处的绝对连续性和渐近行为，推广了自分解分布的几个已知结果。
Multi-type branching processes with immigration generated by point processes
本文探讨了利用点过程框架超越泊松过程的多类型分支过程与非均匀泊松移民的结合方法。
We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramér condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new …
A Galton-Watson branching process with immigration evolving in a random environment is considered. Its associated random walk is assumed to be oscillating. We prove a functional limit theorem in which the process under consideration is normalized by a random coefficient depending on the random environment only. The distribution of the limiting process is described in terms of a strictly stable Levy process and a sequence of independent and identically distributed random variables which is independent of this process.
Exponential ergodicity of branching processes with immigration and competition
本研究证明了带移民和竞争的连续状态分支过程在加权全变差距离下的指数遍历性，适用于次临界、临界和超临界机制，通过构造马尔可夫耦合和非对称控制函数，提供了遍历速率下界的求解方法。
We consider subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are the change of measure and some conditional limit theorems for random walks.
Small positive values and limit theorems for supercritical branching processes with immigration in random environment
本文研究了随机环境中带移民的超临界分支过程的小正值与下偏差不等式，并基于此推导出$\log Z_n$的中心极限定理与Edgeworth展开。
Local extinction in continuous-state branching processes with immigration
本文的目的是观察具有移民的连续状态分支过程（CBI）的零集是无限可分再生集。事实上，它们可以通过Mandelbrot在1972年引入的随机剪切过程来构建。然后，我们展示了如何非常精确的信息可以获得的分支和移民机制的CBI的零集。
Asymptotic behavior of critical indecomposable multi-type branching processes with immigration
本文讨论了当后代均值矩阵不可约时，即当过程不可分解时，一类带移民的临界多型分支过程的渐近性态。证明了临界不可分解多型分支过程的周期序列所构成的适当尺度的随机阶梯函数序列弱收敛于由后代均值矩阵的Perron向量所确定的射线所支持的过程.这些类型可以被划分为非空的互不相交的子集（根据类型的通信），使得属于同一子集的坐标过程是同一平方贝塞尔过程的倍数，并且属于不同子集的坐标过程是独立的。
We prove and extend some results stated by Mark Pinsky: Limit theorems for continuous state branching processes with immigration [Bull. Amer. Math. Soc. 78(1972), 242--244]. Consider a continuous-state branching process with immigration $(Y_t,t\geq 0)$ with branching mechanism $Ψ$ and immigration mechanism $Φ$ (CBI$(Ψ,Φ)$ for short). We shed some light on two different asymptotic regimes occurring when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$ or $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du=\infty$. We first observe that when $\int_{0}\frac{Φ(u)}{|Ψ(u)|}du<\infty$, supercritical CBIs have a growth rate dictate…
A population model with non-neutral mutations using branching processes with immigration
考虑了具有非中性突变的随机种群的平稳连续模型，模型中的状态分支过程具有非齐次迁移。我们假设移民的类型（或突变）是随机的，由恒定的突变率度量给出。我们确定这个过程的一些系谱性质，如：最近的共同祖先（MRCA）的时间分布，瓶颈效应的时间MRCA（这可能是激烈的一些突变率措施），有利的类型的MRCA，祖先的数量的渐近。
A Lamperti-type representation of continuous-state branching processes with immigration
由宽度优先行走的有根树和其序列的世代大小之间的关系的指导下，我们能够包括移民的连续状态分支过程的Lamperti表示。通过求解由一对独立的Levy过程驱动的随机常微分方程，给出了带移民的连续状态分支过程的一种表示.稳定性的解决方案进行了研究，并给出，特别是，极限定理（的类型以前研究的Grimvall，Kawazu和渡边和李）和模拟计划的连续状态分支过程移民。我们进一步应用我们的稳定性分析扩展Pitman的极限定理的条件下的总总体规模的后代法律的Galton-Watson过程。
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…
We consider a continuous-time symmetric branching random walk on multidimensional lattices with immigration and infinite number of initial particles. We assume that at every lattice point a process of birth and death of particles is described by a Bienayme-Galton- Watson branching process. The assumption on immigration of particles is that a new particle can appear at every lattice point from the outside. The subject of the study is a limit distribution of the particle field on the lattice. The differential equations for correlation functions for the number of particles at an arbitrary time mo…
On the hitting times of continuous-state branching processes with immigration
研究了具有移民的连续态分支过程及其在此时刻停止的原态的常能级的首中时的二维联合分布。我们给出了它的拉普拉斯变换的显式表达式。利用这个公式，我们研究了零的极性，并给出了一个瞬时性或常返性的充分必要判据。我们遵循的方法志贺，T。（1990）[A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type.概率论and Related Fields，85（4），425-447]，通过找到生成器的一些$λ$不变函数。
We consider a branching random walk with immigration in a random environment, where the environment is a stationary and ergodic sequence indexed by time. We focus on the asymptotic properties of the sequence of measures $(Z_n)$ that count the number of particles of generation $n$ located in a Borel set of real line. In the present work, a series of limit theorems related to the above counting measures are established, including a central limit theorem, a moderate deviation principle and a large deviation result as well as a convergence theorem of the free energy.
A note on multitype branching processes with immigration in a random environment
我们考虑了一个多型分支过程移民在随机环境中引入的关键[安。15（1987）344- 353]。这是关键表明，根据在[Ann. Probab. 15（1987）344- 353]，分支过程在收敛于一个适当的极限定律的意义上是次临界的。我们补充这个结果的强大数律和中心极限定理的部分和的过程。此外，我们研究的分支过程，也就是说，在连续的时间之间的随机段的振荡的渐近行为时，灭绝发生和过程再次开始与下一波的移民。
We consider a fixed-point equation for a non-negative integer-valued random variable, that appears in branching processes with state-independent immigration. A similar equation appears in the analysis of a single-server queue with a homogeneous Poisson input, feedback and permanent customer(s). It is known that the solution to this equation uniquely exists under mild first and logarithmic moments conditions. We find further the tail asymptotics of the distribution of the solution when the immigration size and branch size distributions are heavy-tailed. We assume that the distributions of inter…
We develop a new methodology for the fluctuation theory of continuous-time skip-free Markov chains, extending the recent work of Choi and Patie [5] for discrete-time skip-free Markov chains. As the main application we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classical family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes.
Branching processes in varying environment with generation-dependent immigration
该研究针对具有世代依赖移民的变化环境分支过程，给出种群必然灭绝的充要条件，并证明临界后代分布且移民均值为正的过程经归一化后渐近分布属于伽马分布族。
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 $α= 1$.
This article studies the stability of solutions of equilibrium equations arising in so-called resource dependent branching processes. We argue that these new models, building on the model already presented by Bruss (1984 a), refined and elaborated in Bruss and Duerinckx (2015) and now extended to allow immigration, are suitable to cope with specific properties of human populations. Our main interest is here to understand under which conditions immigration may lead to an equilibrium. At the same time, we would like to advertize resource dependent branching processes as possibly the best models …
Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration
该研究证明整数集上两类相互作用粒子系统为Pfaffian点过程，经扩散重标度得到实数集上的多种极限Pfaffian点过程，含布朗网对应点集过程。
Cell contamination and branching process in random environment with immigration
我们考虑一个分裂细胞群被寄生虫感染的分支模型。每一个细胞通过从其母细胞遗传和来自群体外部的独立污染来接收寄生虫。寄生虫在细胞内随机繁殖，当细胞分裂时，寄生虫在两个子细胞之间随机繁殖。污染一个给定细胞的寄生虫数量的规律只取决于该细胞是否已经被感染。我们确定了在一个细胞系中寄生虫数量的渐近行为，它遵循一个分支过程在随机环境中与状态相关的移民。然后，我们推导出一个大数定律的渐近比例的细胞与给定数量的寄生虫。主要工具是随机环境中的分支过程和马尔可夫树的大数定律。
On estimation of the convergence rate to invariant measures in markov branching processes with possibly infinite variance and allowing immigration
本文研究了允许移民的连续时间马尔可夫分支过程在临界情形下，子代二阶矩和移民一阶矩可能无穷时，证明其转移函数向不变测度收敛并推导出收敛速度。
Path-valued branching processes and nonlocal branching superprocesses
构造了一族带移民的连续状态分支过程作为时空噪声驱动的随机方程组的解流。该族可以看作是一个带移民的非齐次增路值分支过程。两个非局部分支移民超过程可以从流定义。我们明确确定这些过程的分支和移民机制。这些结果为Aldous和Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34（1998）637-686]和亚伯拉罕和Delmas [安。40（2012）1167-1211]。
On a class of critical Markov branching processes with non-homogeneous Poisson immigration
本文研究带非齐次泊松迁入且后代分布方差无穷的临界马尔可夫分支过程，在迁入均值无穷和泊松强度收敛为常数的条件下，得到了不访问零点的概率渐近行为，并在适当规范化下证明了依赖于后代分布与迁入分布的极限分布。
We consider a branching random walk on $d$-dimensional real space with immigration in a time-dependent random environment. Let $Z_n(\mathbf t)$ be the so-called partition function of the process, namely, the moment generating function of the counting measure describing the dispersion of individuals at time $n$. For $\mathbf t$ fixed, the logarithm $\log Z_n(\mathbf t)$ satisfies a central limit theorem. By studying the logarithmic moments of the intrinsic submartingale of the system and its convergence rates, we establish the uniform and non-uniform Berry-Esseen bounds corresponding to the cen…
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