Limit theorems for sums determined by branching and other exponentially growing processes
Limit theorems for sums determined by branching and other exponentially growing processes
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DOI：
10.1016/0304-4149(84)90311-9
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发表时间：
1984-05
影响因子：
1.4
通讯作者：
P. Jagers;O. Nerman
中科院分区：
文献类型：
--
作者：
P. Jagers;O. Nerman
以随机特征计算的分枝过程被定义为在分枝种群的个体的实际年龄处估计的个体随机过程的叠加过程。现在考虑的特征不仅取决于年龄，还取决于绝对时间。对于超临界过程，证明了一个分布极限定理，这意味着经典的特征和极限定理转化为由这些特征计算的分支过程的极限定理。一个要点是，尽管不同个体的特征应该是独立的，但个体的特征很可能与后者的繁殖相互作用。结果要求1⩽p⩽2具有一定的Lp连续性，证明了它适用于比分支过程更广泛的一类过程.对于Casep=1，给出了若干Poisson型极限和Forp=2的正态逼近.例如，对罕见事件、年龄最大的个体的年龄和人口预测误差的过程获得了结果。这项工作得到了瑞典自然科学研究理事会的资助。
A branching process counted by a random characteristic has been defined as a process which at timetis the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort ofLp-continuity for some 1 ⩽p⩽ 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the casep= 1 a number of Poisson type limits follow and forp= 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.