Controlled Galton-Watson process and its asymptotic behavior

T. Fujimagari
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引用次数: 29

Abstract

1. In a stochastic population process described as a Galton-Watson process each individual splits independently according to a given probability law and new born particles constitute the following generation. In addition to the independence in splitting the law of splitting of each individual depends on neither the generation to which an individual belongs nor the existence of the other individuals of the same generation and is common to all individuals. We shall consider a somewhat generalized Galton-Watson process in the sense that the law of splitting of each individual depends on the total number of individuals of the same generation and the other independence properties are reserved. The object of this note is to study asymptotic behaviors of such processes, although we can hardly obtain any complete results up to now except some partial results. The difficulties in analysing the process will be due to the dependence introduced above from which it no longer holds such as the iteration property of a generating function which plays a fundamental role in GaltonWatson processes. We shall formulate the process under consideration as follows. Let Zn be the size or the total number of individuals which belong to the n-th generation and given a sequence of probability distributions £>(i)= \pr(i): rΞ>0}, z=0, 1,2, ••• oo
可控Galton-Watson过程及其渐近性
1. 在被描述为高尔顿-沃森过程的随机种群过程中,每个个体根据给定的概率定律独立分裂,新出生的粒子构成下一代。除了分裂的独立性之外,每个个体的分裂规律既不取决于个体所属的时代,也不取决于同一时代其他个体的存在,而是所有个体所共有的。我们将考虑一个有点广义的高尔顿-沃森过程,在这个意义上,每个个体的分裂定律取决于同一代个体的总数,并且保留其他独立性。本文的目的是研究这类过程的渐近行为,尽管到目前为止,除了一些部分结果外,我们几乎没有得到任何完整的结果。分析过程的困难将是由于上面介绍的依赖性,它不再具有诸如在高尔顿-沃森过程中起基本作用的生成函数的迭代性质。我们将把审议中的程序拟订如下。设Zn为属于第n代的个体的大小或总数,给定一个概率分布序列£>(i)= \pr(i): rΞ>0}, z= 0,1,2,••••oo
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