一类上临界带移民分支过程的下偏差概率估计

对于后代分布为{p_i, i≥0}的上临界带移民分支过程{Z_n},如果分支和移民分布满足适当的矩条件,则Z_n/mⁿ几乎处处收敛到某个非退化的极限,其中■为过程后代分布的均值.本文给出了p₀> 0时该过程下偏差概率P(Z_n=k)的渐近行为,其中k∈[k_n, mⁿ], k_n→∞(n→∞),这一结果可作为文献[8]中Schr?der情形结论的补充.     

测度值分枝过程与移民过程

本文介绍了测度值分枝过程和由斜卷积半群定义的伴随移民过程的基本理论和研究现状,主要内容包括：分枝粒子系统的收敛;超过程的基本正则性和极限定理;非线性微分方程;广义分枝模型;斜卷积半群和进入律;用Ｋｕｚｎｅｔｓｏｖ过程构造移民过程等。     

带移民Jirina过程的弱极限定理

该文在矩条件下讨论了一列带移民Jirina过程的弱极限定理.按照极限过程的不同对矩条件作了简单分类.文章证明了在不同的矩条件下,一列带移民Jirina过程适当规范后可以在Skorokhod空间分别弱收敛到连续分支过程,带移民的连续分支过程,不连续的带移民分支过程以及确定性过程.对最后这种情形,还给出了一个波动极限定理.     

带移民的马尔可夫分支过程的唯一性

本文考虑了一类带移民的马尔可夫分支过程,给出了这个过程的唯一性和正则性的判定准则。     

Galton-Watson分支过程的谱半径的概率刻画

谱半径是不可约马尔可夫链的一个很重要的特征数字.Galton-Watson分支过程是一类特殊的马尔可夫链,我们已经证明了在不可约的条件下,Galton-Watson分支过程的谱半径等于其对应的概率母函数f(s)在灭绝概率q点的导数值.本文主要从理论上刻画从过程的任何状态逃离速度的Galton-Watson分支过程的谱半径的概率意义.     

测度值分枝扩散过程在球域上负荷的概率估计

本文通过对一类非线性微分方程解的性质研究，给出了测度值分枝扩散过程在球域上负荷概率的精确估计     

随机环境下两个上临界分支过程的参数比较

设(Z_1,n)_n≥0和(Z_2,n)_n≥0是两个在独立同分布随机环境下的上临界分支过程,并且其关键参数分别为μ₁和μ₂.容易知道,在适当条件下,■和■分别依概率收敛到μ₁和μ₂.该文旨在讨论两个上临界分支过程的关键参数之差μ₁-μ₂的估计问题,它可以被看作是一类双样本U统计量问题.我们得到了■的中心极限定理,非一致性Berry-Esseen估计和Cramér型中偏差.最后,作为应用部分,指出了以上的结果可用于关键参数置信区间的构造.     

上临界迁出分枝过程的收敛速率

本文证明上临界迁出分枝过程的规范化过程的收敛性,并讨论其收敛速率.     

带移民的非临界分枝相依空间运动超过程

通过对一列带正跳跃的超过程取极限,本文构造了带移民的相依空间运动超过程.在此基础上,利用Dawson型的Girsanov变换得到了相应的非临界分枝,此变换同时给出依赖于整体状态的空间漂移.     

带移民的非临界分枝相依空间运动超过程

通过对一列带正跳跃的超过程取极限,本文构造了带移民的相依空间运动超过程.在此基础上,利用 Dawson型的Girsanov变换得到了相应的非临界分枝,此变换同时给出依赖于整体状态的空间漂移.     

Small value probabilities for continuous state branching processes with immigration

We consider the small value probability of supercritical continuous state branching processes with immigration.From Pinsky(1972) it is known that under regularity condition on the branching mechanism and immigration mechanism,the normalized population size converges to a non-degenerate finite and positive limit W as t tends to infinity.We provide sharp estimate on asymptotic behavior of P(W≤ε) as ε→ 0+ by studying the Laplace transform of W.Without immigration,we also give a simpler proof for the small value probability in the non-subordinator case via the prolific backbone decomposition.     

Lower deviations for supercritical branching processes with immigration

For a supercritical branching processes with immigration \{{{\displaystyle Z}}_n\}; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P({{\displaystyle Z}}_n={{\displaystyle k}}_n) with {{\displaystyle k}}_n=o(m^n) as n\to \infty. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.     

Stochastic equations for two-type continuous-state branching processes with immigration

A two-dimensional stochastic integral equation system with jumps is studied. We first prove its unique weak solution is a two-type continuous-state branching process with immigration. Then the comparison property of the solution is established. These results imply the existence and uniqueness of the strong solution of the stochastic equation system.     

Nonlinear branching processes with immigration

The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.     

Existence,uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.     

Parameter estimation in two-type continuous-state branching processes with immigration

We study the parameter estimation of two-type continuous-state branching processes with immigration based on low frequency observations at equidistant time points. The ergodicity of the processes is proved. The estimators are based on the minimization of a sum of squared deviation about conditional expectations. We also establish the strong consistency and central limit theorems of the conditional least squares estimators and the weighted conditional least squares estimators of the drift and diffusion coefficients based on low frequency observations.