Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes''
P. E. Ney, Anand N. Vidyashankar
"局部极限理论和超临界分支过程的大偏差“的更正和确认[数学PR/0407059]
Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes'' [math.PR/0407059]
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Supercritical branching diffusions in random environment
已知在恒定环境中以最终灭绝为条件的超临界分支过程是亚临界分支过程。随机环境的情况更加微妙。以种群最终灭绝为条件的随机环境中的超临界分支扩散（BDRE）不是BDRE。而以最终绝灭为条件的超临界BDRE的总体规模的猝灭规律与亚临界BDRE的总体规模的猝灭规律是一致的。因此，超临界BDRE具有类似于在随机环境中的亚临界支化过程的熟知相变的相变。
In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.
Central Limit Theorems for Supercritical Branching Nonsymmetric Markov Processes
本文对一类不一定对称的超临界分支马氏过程建立了一个空间中心极限定理，该过程的分支机制是空间依赖的，且满足二阶矩条件.这个中心极限定理推广和统一了最近在\cite{RSZ2}中得到的关于超临界分支对称马氏过程的中心极限定理。为了证明我们的中心极限定理，我们必须仔细地发展非对称强连续半群的谱理论，这应该是独立的兴趣。
Central Limit Theorems for Supercritical Branching Markov Processes
本文对一类具有一般空间依赖分支机制的超临界分支马氏过程建立了空间中心极限定理。这些定理是文献[1]中证明的具有二元分枝机制的分枝过程的空间中心极限定理的推广。与文[1]的结果相比，我们的中心极限定理在正态随机变量是非退化的意义上更令人满意.
Limit theorems for supercritical branching processes in Markovian environment with linear fractional offspring distributions
研究马尔可夫环境中具线性分数子代分布的超临界分支过程渐近行为，通过考察强及中间强超临界情形发现过程行为的相变。
$L\log L$ condition for supercritical branching Hunt processes
本文利用脊分解和测度的鞅变换建立了分支Hunt过程的Kesten-Stigum $L\log L$定理。这个结果是Asmussen-Hering（1976）和Hering（1978）中关于分支扩散的结果的推广。
Let $\left \{ Z_{n}, n\ge 0 \right \}$ be a supercritical branching process in an independent and identically distributed random environment $ξ=\left ( ξ_{n} \right )_{n\geq 0} $. In this paper, we get some deviation inequalities for $\ln \left (Z_{n+n_{0} } / Z_{n_{0} } \right ).$ And some applications are given for constructing confidence intervals.
We study supercritical branching processes under the influence of an i.i.d. emigration component. We provide conditions, under which the lifetime of the process is finite respectively has a finite expectation. A new version of the Kesten-Stigum theorem is obtained and the extinction probability for a large initial population size is related to the tail behaviour of the emigration.
Decomposition of supercritical linear-fractional branching processes
超临界单型Bienyamé-Galton-Watson过程可以看作是由两类粒子构成的可分解的分支过程：一类是具有无限下降线的粒子，另一类是具有有限后代数的粒子.本文分析了具有可数多个类型的线性分式Bienyamé-Galton-Watson过程的分解。
Supercritical general branching processes conditioned on extinction are subcritical
众所周知，一个简单的超临界Bienaymé-Galton-Watson过程，如果条件消失，就会变成一个亚临界过程。我们证明，相应的适用于一般的，多类型的分支，在那里生育可能发生在不同的年龄，寿命可能取决于繁殖，整个过程的事件，从而受到制约灭绝。
We consider a supercritical symmetric continuous-time branching random walk on a multidimensional lattice with a finite number of particle generation sources of varying positive intensities without any restrictions on the variance of jumps of the underlying random walk. It is assumed that the spectrum of the evolution operator contains at least one positive eigenvalue. We prove that under these conditions the largest eigenvalue of the evolution operator is simple and determines the rate of exponential growth of particle quantities at every point on the lattice as well as on the lattice as a wh…
Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions
在本文中，我们刻画了随机环境中的超临界分支过程的行为与线性分数后代分布，条件是有小，但在一些大的代正值。正如在以前的工作中已经注意到的那样，在过程的行为中存在相变。在这里，我们研究了强和中等超临界制度的主要结果是一个条件极限定理的重新标度相关的随机游动在中等情况下。
A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein-Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment mea…
Limiting laws of supercritical branching random walks
在这篇文章中，我们明确的重整化超临界分支随机游动的法律，给予信贷的作者在以前的作品中制定的一个猜想连续模拟的分支随机游动。此外，在齐次树上的分支随机游动的情况下，我们表达了相应的极限重正化吉布斯测度的定律，在这个离散模型中，证实了物理学家关于极限跳跃的泊松-狄利克雷性质的理论，并通过给出这些跳跃的空间分布来精确化猜想。
Limit theorems for the empirical distribution of supercritical branching random walks on transitive graphs
本文研究了传递图上超临界分支随机游走的经验分布，证明了大数定律和Stam型中心极限定理，解决了Kaimanovich-Woess提出的问题。
Exponential rate of L_p-convergence of intrinsic martingales in supercritical branching random walks
设$W_n, n\in\mn_{0}$是超临界分支随机游动中具有几乎必然极限$W$的内禀鞅.我们提供的$L_p$ -收敛的系列$\sum_{n\ge 0} e^{an}(W-W_n)$为$p>1$和$a>0$的标准。该结果可以看作是关于$\me |W-W_n|^p$收敛到零的指数速率的陈述。
Small value probabilities of additive and derivative martingales in supercritical branching Brownian motions and super Brownian motions
该研究建立了超临界分支布朗运动和超布朗运动中两类鞅小值概率的渐近结果，推广了相关分支过程的已有结论，且与已有文献结果吻合。
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ= (ξ_n)$. We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for $\log Z_n$ under the annealed law $\mathbb P$. We also improve some earlier results about the harmonic moments of the limit variable $W=lim_{n\to \infty} W_n$, where $W_n =Z_n/ \mathbb{E}_ξ Z_n$ is the normalized population size.
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。
Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $ξ$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j | Z_0 = k)$ as $n \to \infty$, for each $j \geq k,$ assuming that $\mathbb{P} (Z_1 = 0) =0$. We also determine the critical value for the existence of harmonic moments of the random variable $W=\lim_{n\to\infty}\frac{Z_n}{\mathbb E (Z_n|ξ)}$ under a simple moment condition.
Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for $\displaystyle\mathbb{P}\left[\frac{\log Z_n}{Mn}-μ\geq x\right]$ for any $x\geq3$ are obtained, by applying an extension of the Hoeffding type inequalities.
Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching
研究超临界分支布朗运动中前沿附近粒子分布的长期行为，利用抛物型PDE解的渐进行为证明粒子数收敛性及极限分布依赖域位置。
We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgew…
Exponential rate of almost sure convergence of intrinsic martingales in supercritical branching random walks
给出了超临界分支随机游动的内禀鞅指数快速收敛到极限的充分条件。Galton-Watson过程的情况下，特别是包括，使我们的结果可以被看作是一个结果的推广，在经典的论文中给出的Asmussen和Hering。作为一个辅助工具，我们证明了最终版本的两个结果的指数更新措施，这可能是有趣的本身，并纠正，推广和简化一些早期的工作。
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。
Haldane's asymptotics for supercritical branching processes in an iid random environment
本文研究了iid随机环境中轻微超临界分支过程的存活概率渐近衰减，证明了在远离临界状态时经典Haldane渐近性依然成立，并通过永续年金极限定理提供理论支撑。
A moment approach to the law of large numbers for supercritical branching Markov processes
本文通过矩方法为超临界分支马尔可夫过程提供新的大数定律证明，并证明极限变量分布由其矩完全确定。
Let $(W_n(θ))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(θ)$ its limit. Assuming essentially that the martingale $(W_n(2θ))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(θ)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(θ) - W_{n+r}(θ))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(θ)-W_n(θ)$, as $n\to\infty$.
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…
Limit theorems for supercritical remaining-lifetime age-structured branching processes
本文研究超临界年龄结构分支模型，给出马尔萨斯归一化随机测度收敛的充要条件，并在特定条件下加强极限理论，证明随机归一化的中心极限定理。
Consider a branching system with particles moving according to an Ornstein-Uhlenbeck process with drift $μ>0$ and branching according to a law in the domain of attraction of the $(1+β)$-stable distribution. The mean of the branching law is strictly larger than $1$ implying that the system is supercritical and the total number of particles grows exponentially at some rate $λ>0$. It is known that the system obeys a law of large numbers. In the paper we study its rate of convergence. We discover an interesting interplay between the branching rate $λ$ and the drift parameter $μ$. There are three r…
We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman's martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the …
The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures' laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its ma…
Convergence of complex martingales in supercritical multi-type general branching processes in $L^q$ for $1 < q \leq 2$
该研究针对有限类型的超临界一般分支过程，构造了类似单类型过程中存在的复值Nerman型鞅，并给出了这类鞅在$L^q$（$1<q\leq2$）下收敛的充分条件，拓展了多类型分支过程的鞅收敛理论。
Local limit theory and large deviations for supercritical Branching processes
本文研究了超临界Galton-Watson过程{Z_n：n \ge 1}增长的几个方面，提出了由Schroder常数决定的临界现象。本文发展了Z_n的局部极限理论，即P（Z_n=v_n）作为v_PDMATHABMN \infty的行为，并利用它研究了{ Y_{Z_n}：nPDMATHABMM 1}的条件大偏差，其中Y_n满足LDP，特别是{Z_n^{-1} Z_{n+1}：n \ge 1}在Z_n \ge v_n上的条件.
Let $\{Z_n, n\geq 0\}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for $\ln (Z_{n+n_0}/Z_{n_0})$ % under the annealed law, uniformly in $n_0 \in \mathbb{N}$, which extend the corresponding results by Grama et al. (Stochastic Process.\ Appl. 2017) established for $n_0=0$. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. (2017) are relaxed in our present setting. An example of appl…
Consider a branching random walk on the real line. Madaule showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probab…
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…
Inverting the operation of conditioning a branching process on extinction
该研究针对分支过程的灭绝条件化操作提出逆问题，证明温和假设下共轭分支过程存在且多类型时不唯一，其结果可用于随机平面映射构造与癌症建模。
From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment
本文基于鞅极限理论最新进展，为独立同分布环境下的超临界分支过程$(\log Z_n)_{n\geq0}$建立了重对数律、强大数律、不变性原理及Zolotarev和Wasserstein距离下的中心极限定理最优收敛速率。
Maximal and minimal displacement of supercritical branching random walks on free products of groups
研究超临界分支随机游走于群自由积上的最大和最小位移增长特性，证明其几乎必然线性增长，通过大偏差率函数与水平线的交点确定速度。
Law of iterated logarithm for supercritical non-symmetric branching Markov process
本文研究了超临界非对称分支马尔可夫过程在不同矩条件下的迭代对数定律，探讨了分支机制与函数类对结果的影响。
In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (a.k.a. Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value $W$ figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman's martingale, converges to its limit $W$. More precisely, assuming the existence of a Malthusian parameter $α> 0$ and $W_0\in L^2$, we prove a functional c…
A formula for the periodic multiplier in left tail asymptotics for supercritical branching processes in the Schröder case
针对Schröder情形下超临界分支过程左尾渐近的幂律乘周期函数特性，研究给出了其中乘法周期函数的显式表达式。
Law of the iterated logarithm for supercritical non-local spatial branching processes
研究超临界非局部空间分支过程相关鞅的几乎必然行为，在四阶矩条件下建立重对数律型极限定理，发现三种情形，得到线性泛函重对数律，改进分支马尔可夫过程相关文献结果，给出超过程新结果。
A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks
无穷多个独立同分布被乘性随机游动折扣的随机变量称为不变量，并且已经被许多作者研究过。本文给出了这类随机变量在极小条件下的对数型矩结果，并利用它研究了a.s.与超临界分支随机游动相关的某些鞅的极限。第二作者在[Iksanov，A.M.（2004）.具有无穷多个和数的BRW光滑变换的初等不动点。Stoch. Proc.Appl.114，27-50.]，在考虑最初由Lyons在[Lyons，R.（1997年）。分支随机游动Biggins鞅收敛的一条简单途径。在K.B.的Athreya，Jagers，P.（编）. Classical and Modern Branching Processes，IMA，Mathematics and its Applications，vol.84，Springer，Berlin，217-221.]。在最一般的情形下，我们也给出了这些鞅一致可积的充要条件，这特别意味着经典的（LlogL）-条件并不总是必要的.
Approximate Modeling for Supercritical Galton-Watson Branching Processes with Compound Poisson-Gamma Distribution
本文研究了超临界分支过程在平均子代分布接近 1 的渐进行为下的种群大小分布，证明其可近似为复合泊松-伽马分布，并通过实验验证了该模型在电子倍增器等级联乘法过程中的有效性。
We give necessary and sufficient conditions for laws of large numbers to hold in $L^2$ for the empirical measure of a large class of branching Markov processes, including $λ$-positive systems but also some $λ$-transient ones, such as the branching Brownian motion with drift and absorption at $0$. This is a significant improvement over previous results on this matter, which had only dealt so far with $λ$-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly p…
Small positive values for supercritical branching processes in random environment
随机环境中的分枝过程（BPREs）$(Z_n:n\geq0)$是Galton-Watson过程的推广，其中在每一代中，再生律在i.i.d.中随机选取。方式在超临界情况下，过程以正概率存活，然后几乎肯定以几何级数增长。本文主要研究过程在大时间内取正值但值较小的罕见事件。我们把$\mathbb{P}(1 \leq Z_n \leq k|Z_0=i)$，$k,i\in\mathbb{N}$的渐近行为描述为$n\rightarrow \infty$ .更准确地说，我们的特点是指数下降的$\mathbb{P}(Z_n=k|Z_0=i)$使用脊柱表示由于盖革。然后，我们提供了一些界限，这种减少率。如果再生律是线性分式的，则该速率变得更加明确，并且出现两种状态。此外，我们表明，这些制度影响最近的共同祖先的渐近行为时，人口的条件是小，但积极的大时代。
Branching Processes in Random Environment (BPREs) $(Z\_n:n\geq0)$ are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of $Z$, which means the asymptotic behavior of the probability $\{1 \leq Z\_n \leq \exp(nθ)\}$ as $n\rightarrow \in…
In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state $E$ is a finite set and the underline motion is an irreducible Markov chain on $E$, our results are superprocess analogs of the functional central limit theorems of \cite{Janson} for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in \cite{RSZ3}.
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…
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。
Stable convergence of conditional least squares estimators for supercritical continuous state and continuous time branching processes with immigration
本文证明了基于离散时间观测的带移民超临界连续状态与连续时间分支过程漂移参数的条件最小二乘估计量具有稳定收敛性。
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for…
In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Miłoś (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is that it allows us to characterize the limit Gaussian field. In the case of supe…
We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amou…
Strong law of large numbers and L log L condition for supercritical branching processes
研究结构化种群的超临界分支过程，在重正化半群加权全变差范数收敛假设下，结合鞅族、半群收缩等技术，证明归一化经验测度强收敛等，在Llog L条件下扩展大数定律适用范围。
Large deviation probabilities for the range of a d-dimensional supercritical branching random walk
本文研究d维超临界分支随机游走的值域，证明R_n/n依概率收敛于正常数x*，并给出其下、上偏差概率的衰减速率，同时证实Engländer猜想。
Asymptotic fluctuations in supercritical multi-type Crump-Mode-Jagers processes
研究超临界多类型Crump-Mode-Jagers过程的渐近波动，在特定二阶矩假设下证明其中心极限定理，扩展了单类型中心极限定理并统一了特定分支过程的极限结果。
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