Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.
Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.
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…
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。
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
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.
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.
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).
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…
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 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.
Multi-type branching processes with immigration generated by point processes
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学克利门特·奥赫里德斯基数学与信息学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息研究所）
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.
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.
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…
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.
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.
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
