【作者】 王洁；

【导师】 李应求；

【作者基本信息】 长沙理工大学 ， 概率论和数理统计， 2009， 硕士

【摘要】 1873年Galton和Waston分别就他们在探讨英国贵族姓氏继承与谱系消亡问题中建立起来的一种新的随机过程模型发表了论文,正是这两篇论文奠定了经典分枝过程的理论基础.分枝过程是概率理论研究中活跃而富有成果的分支之一.离散时间离散状态的Galton-Waston过程(GW-过程)是最简单的分枝过程,它是描述粒子群体发展演化的数学模型,经典分枝过程的基本假定是每个粒子按照相同的概率分布彼此独立的进行繁衍,而在实际问题中,个体间繁衍后代要受到多种因素的相互作用,由此引入了各类分枝模型,从齐次的分枝过程到非齐次的分枝过程,从单型的分枝过程到多型的分枝过程,从经典的分枝过程到随机环境中的分枝过程等.在众多模型的扩展中,从确定环境中的受控分枝过程到随机环境中的受控分枝过程的拓展研究是近十年来广为关注的一个课题,多位学者对它进行了深入的研究.从确定环境中的分枝过程到随机环境中的分枝过程的推广研究已经拥有了一个庞大完整的理论体系,此体系主要从四个大的方面进行了深入的创新研究,我们以此推广模式为基础,亦从四个方面将确定环境中的受控分枝过程的研究理论推广到随机环境中的受控分枝过程.首先,把确定环境中的受控分枝过程模型拓展到随机环境中的受控分枝过程模型;其次,以确定环境中的受控分枝过程相关概率母函数间关系及相关矩量性质为前提,讨论随机环境中的受控分枝过程概率母函数间关系及各矩量间关系,再次,探讨随机环境中受控分枝过程的灭绝概率问题;最后,考虑随机环境中受控分枝过程的渐近性质问题.更多还原

【Abstract】 In 1873 Galton and Waston published parpers respectively about the new random process model established in the issue on surname inheritance and lineage extinction of the British aristocracy.It is just these two papers laid the theoretical foundation for the classic branching processes. Research on branching processes is one active and fruitful branch in the researching field of probability theory. Galton-Waston process (GW-process) with discrete states in discretetime is the simplest model of branching processes which describe the evolution of particle groups and has basic assumption each particle produces descendant independently with the same probability distribution. But in reality, the reproduction between the individuals is affected by many factors, so we introduced various types of branching process model, from homogeneous branching processes to non-homoge- neous branching process, from single type branching processes to multi- type branching processes, from classical branching processes to branching processes in random environments.Among the many expansion models, expansion from the controlled branching processes in determinate environment to the controlled branching processes in random environment is a widespread concerned topic over the past decade that a number of scholars had an in-depth research on.The research on the expansion from branching processes in determinate environment to branching processes in random environment has a large complete theoretical system which conducts an in-depth study of innovation in four major areas. We will generalize the controlled branching processes model from the determinate environment to random environment basing on the above-mentioned four aspects. Firstly, the model of the controlled branching processes in determinate environment is extended to the model of controlled branching processes in random environment. Secondly, the relationships between the probability generating function and the related moment about controlled branching processes in random environment are considered acoording to the relationships in the determinate environment. Thirdly extinction probability of the controlled branching processes in random environment is discussed. Finally the asymptotic properties is studied.更多还原