Péter Kevei, Péter Wiandt
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.
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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.
For multitype branching processes with immigration evolving in a random environment and producing a final product we find the tail distribution of the size of the final product accumulated in the system for a life period. Using this result we investigate the tail distribution of the busy periods of the branching type polling systems with random service disciplines and random positive switch-over times
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.
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.
In this work, we study asymptotics of multitype Galton-Watson trees with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analogue of a result obtained by Miermont in the case of multitype Galton-Watson trees with finite covariance matrices of the offsprin…
We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some suitable moment assumptions.
We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching pro…
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。
In this paper, we first form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the hitting time of corresponding random walks.
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.
In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classi…
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.
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…
Critical exponential tiltings for size-conditioned multitype Bienaymé--Galton--Watson trees
本文研究了在顶点数量满足线性关系条件下的多类型 Bienaymé--Galton--Watson 树，利用临界指数倾斜方法证明了存在等价的临界后代分布，并据此刻画了树尺寸趋于无穷大时的局部极限。
We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.
We consider a multitype branching process with immigration in a random environment introduced by Key in [Ann. Probab. 15 (1987) 344--353]. It was shown by Key that, under the assumptions made in [Ann. Probab. 15 (1987) 344--353], the branching process is subcritical in the sense that it converges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior of oscillations of the branching process, that is, of the random segments between successive times when the e…
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.
Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated $Q$-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the $Q$-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained $Q$-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype…
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。
Existence of critical tiltings and local limits of general size-conditioned Bienaymé-Galton-Watson multitype trees
本文研究多类型 BGW 树在类型计数线性组合条件下的结构，通过子代分布指数倾斜的几何分析构造同条件分布的临界树，从而广泛证明非临界情形局部极限的存在性。
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.
Multitype Lévy trees as scaling limits of multitype Bienaymé-Galton-Watson trees
研究通过特定条件对多型比内亚梅-加尔顿-沃茨树进行缩放，得到多型莱维树的极限结构，提出了一种基于莱维场的不变原理，扩展了单型连续随机树的研究。
Fractal random sets associated with multitype Galton-Watson trees
该研究用多类型 Galton-Watson 树编码由正则镶嵌相邻瓦片迭代生成的随机平面集，证明极限集的盒维数与 Hausdorff 维数相等，并由再生矩阵谱半径计算，还给出三类镶嵌示例。
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 …
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 $α= 1$.
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…
Let $A$ be a primitive matrix and let $λ$ be its Perron-Frobenius eigenvalue. We give formulas expressing the associated normalized Perron-Frobenius eigenvector as a simple functional of a multitype Galton-Watson process whose mean matrix is $A$, as well as of a multitype branching process with mean matrix $e^{(A-I)t}$. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.
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。
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 somewhat didactic note we give a detailed alternative proof of the known result due to Wei and Winnicki (1989) which states that under second order moment assumptions on the offspring and immigration distributions the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (starting from not necessarily zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki (1989) is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to Ispán…
In the spirit of Duqesne and Winkel (2007) and Berestycki et al. (2011) we show that supercritical continuous-state branching process with a general branching mechanism and general immigration mechanism is equal in law to a continuous-time Galton Watson process with immigration with Poissonian dressing. The result also characterises the limiting backbone decomposition which is predictable from the work on consistent growth of Galton-Watson trees with immigration in Cao and Winkel (2010).
Sampling schemes of multitype continuous-time Bienaymé-Galton-Watson trees and limiting critical genealogies
研究了在连续时间多型比内梅-加顿-沃森过程中，固定时间点上抽取k个区分粒子的两种类型依赖采样方案下的谱系结构，分析了最近公共祖先时间、祖先子代分布及类型依赖祖先结构，并在临界条件下证明了极限谱系的鲁棒性。
In this article, we prove a joint large deviation principle in $n$ for the \emph{empirical pair measure} and \emph{ empirical offspring measure} of critical multitype Galton-Watson trees conditioned to have exactly $n$ vertices in the weak topology. From this result we extend the large deviation principle for the empirical pair measures of Markov chains on simply generated trees to cover offspring laws which are not treated by \cite[Theorem~2.1]{DMS03}. For the case where the offspring law of the tree is a geometric distribution with parameter \sfrac{1}{2}, we get an exact rate function. All o…
Uniform sampling of multitype continuous-time Bienaymé-Galton-Watson trees
研究连续时间多类型Bienaymé-Galton-Watson过程在固定时间点均匀无放回抽取k个个体的谱系，通过k-脊分解和测度变换，揭示类型间复杂交互，为后续研究奠定基础。
A multitype Galton-Watson model for rejuvenating cells
本文采用多类型 Galton-Watson 过程对分裂细胞群体建模，以有害蛋白质数量定义细胞生物学年龄，研究细胞分裂时因蛋白质转移至子细胞而发生的复壮现象，并分析了预期复制寿命、群体增长率等生物学特征。
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 …
We study a genealogical model for continuous-state branching processes with immigration with a (sub)critical branching mechanism. This model allows the immigrants to be on the same line of descent. The corresponding family tree is an ordered rooted continuum random tree with a single infinite end defined thanks to two continuous processes denoted by $(\overleftarrow{H}_t ;t\geq 0)$ and $(\overrightarrow{H}_t ;t\geq 0)$ that code the parts at resp. the left and the right hand of the infinite line of descent of the tree. These processes are called the left and the right height processes. We defi…
The paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions.
We describe the asymptotic behavior of the conditional least squares estimator of the offspring mean for subcritical strongly stationary Galton--Watson processes with regularly varying immigration with tail index $α\in (1,2)$. The limit law is the ratio of two dependent stable random variables with indices $α/2$ and $2α/3$, respectively, and it has a continuously differentiable density function. We use point process technique in the proofs.
We prove a general fluctuation limit theorem for Galton-Watson branching processes with immigration. The limit is a time-inhomogeneous OU type process driven by a spectrally positive Levy process. As applications of this result, we obtain some asymptotic estimates for the conditional least-squares estimator of the offspring means and variances of the offspring and immigration distributions.
Fast Simulation of Size-Constrained Multitype Bienaymé-Galton-Watson Forests and Applications
本文提出一种高效算法，用于均匀生成给定度序列的多类型森林，扩展了单类型构造，并应用于条件多类型Bienaymé-Galton-Watson森林的模拟。
We consider the time evolution of the lattice subcritical Galton-Watson model with immigration. We prove Carleman type estimation for the cumulants in the simple case (binary splitting) and show the existence of a steady state. We also present the formula of the limiting distribution in a particular solvable case.
Exponential Tail Estimates for Multitype Poisson Branching Processes and Application to Hawkes Processes
针对多类型泊松分支过程的总后代线性泛函，建立了显式指数矩界，并应用于多类型霍克斯过程，得到高维情形下的改进结果。
We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton-Watson with immigration processes. We also significantly improve some results on multitype size-dependent Galton-Watson processes.
Large deviation rates for supercritical multitype branching processes with immigration
本文研究带移民的超临界多型分支过程，证明其规范化过程为收敛鞅，并刻画了若干统计量收敛到极限时的大偏差速率，包括几何与超几何衰减情形。
Detection of extremely rare variant alleles, such as tumour DNA, within a complex mixture of DNA molecules is experimentally challenging due to sequencing errors. Barcoding of target DNA molecules in library construction for next-generation sequencing provides a way to identify and bioinformatically remove polymerase induced errors. During the barcoding procedure involving $t$ consecutive PCR cycles, the DNA molecules become barcoded by unique molecular identifiers (UMI). Different library construction protocols utilise different values of $t$. The effect of a larger $t$ and imperfect PCR ampl…
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 a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.
Binary Galton-Watson trees with mutations
本文研究了带有突变和逆转的二元加顿-沃特森树，通过递归计算类型分布来估计长期类型比例和首次出现时间分布。
Critical Multitype Branching Processes with Random Migration
本文提出带随机迁移的临界多类型分支过程，给出无限增长的充分条件，研究规范化后的渐近分布，并建立 Feller 型扩散逼近。
Complete left tail asymptotic for supercritical multitype branching processes
本文推导了超临界多类型Galton-Watson过程鞅极限密度的完全左尾渐近级数，证明其处处收敛，并给出快速计算的近似，为首个多类型分支过程左尾完整结果。
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