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Quadratic transformations: a model for population growth. II

Published online by Cambridge University Press:  01 July 2016

Harry Kesten
Harry Kesten*
Affiliation:
Cornell University
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In this last part the Fn(i) and Mn(i) are considered as random variables whose distributions are described by the model and various mating rules of Section 2. Several convergence results will be proved for those specific mating rules, but we begin with the more general convergence theorem 6.1. The proof of this theorem brings out the basic idea of this section, namely that when Fn and Mn are large, Fn + 1(i) and Mn + 1(i) will, with high probability, be close to a certain function of Fn(·) and Mn(·) (roughly the conditional expectation of Fn+1(i) and Mn + 1(i) given Fn(·) and Mn(·)). As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equality for some transformation T of the form (1.4), (1.5). More generally for fixed k except on a set whose probability is small when Fn and Mn are large. If the theorems of Section 3 or 4 apply, Tk(fn(·), mn(·)) will be close to a fixed vector ζ when k is large and thus there is hope that fn(·) and mn(·) will converge, once Fn and Mn become large. We therefore have to put on some conditions which will make Fn and Mn grow. This is the role of (6.34) and, to some extent, also of (6.17). The main difficulty is that the expected size of the (n + 1)th generation, given the nth generation, depends on the frequencies of the different types present in the nth generation. Even if (6.34) holds, the conditional expected size of the (n + 1)th generation, given the nth generation, may actually be smaller than the size of the nth generation for certain directions fn(·), m(·).

Type
Research Article
Information
Advances in Applied Probability , Volume 2 , Issue 2 , Autumn 1970 , pp. 179 - 228
Copyright
Copyright © Applied Probability Trust 1970 

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