Limit theorems for sums determined by branching and other exponentially growing processes
type: Item
instance of: scholarly article
title: Limit theorems for sums determined by branching and other exponentially growing processes
zbMATH Open document ID: 0532.60081
DOI: 10.1016/0304-4149(84)90311-9
author: P. Jagers
author: Olle Nerman
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: This paper considers the general branching process counted by random characteristics, details of which can be found in the first author's book, Branching processes with biological applications. (1975; Zbl 0356.60039). Each individual x, with birth time \(\sigma_ x\), has associated with it a random function, called a characteristic, \(\chi_ x(u)\). Results on the convergence of \(e^{-\alpha t}\sum_{x}\chi_ x(t-\sigma_ x)\) with \(\alpha\) the Malthusian parameter have, at least for particular choices of \(\chi\), a long history culminating in the paper by the second author, Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 365-395 (1981; Zbl 0451.60078). Here the framework is extended somewhat so that each individual has associated with it a whole collection of characteristics, one for each \(t\geq 0\), \(\{\chi_{t,x}:t\geq 0\}\) and \(\sum_{x}\chi_{t,x}(t- \sigma_ x)\) is considered as \(t\to \infty\). This can be thought of as the row sums of a triangular array of characteristics. Suitable conditions on the collection of characteristics allow the convergence in distribution of \(\sum_{x}\chi_{t,x}(t-\sigma_ x)\) as \(t\to \infty\) to be established. Also, as the authors point out, for the result obtained the important feature of the underlying process leading to the birth times is not that it is a general branching process but that it grows exponentially. A wide variety of cases are covered by the theorem obtained and various special cases are discussed. A number of these relate to various kinds of rare events when the limits obtained are mixed Poisson. Others relate to the random error of population projections when the limits are mixed normal distributions (i.e. normals with a random variance). The main result also holds for the lattice case of the general branching process. For the supercritical Galton-Watson process one of the authors' population projection examples corresponds to a theorem of \textit{C. C. Heyde} on the rate of convergence in this process; see J. A…
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J85
zbMATH DE Number: 3844784
zbMATH Keywords: branching process
zbMATH Keywords: population
zbMATH Keywords: rare event
zbMATH Keywords: prediction
zbMATH Keywords: triangular array of characteristics
zbMATH Keywords: supercritical Galton-Watson process
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90311-9
OpenAlex ID: W1969657280
cites work: Uniformity in weak convergence
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cites work: On the convergence of supercritical general (C-M-J) branching processes
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description: scientific article; zbMATH DE number 3844784