Quadratic transformations: a model for population growth. II

Abstract

In this last part theF_n(i) andM_n(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 whenF_nandM_nare large,F_{n + 1}(i) andM_{n + 1}(i) will, with high probability, be close to a certain function ofF_n(·) andM_n(·) (roughly the conditional expectation ofF_n+1(i) andM_{n + 1}(i) givenF_n(·) andM_n(·)). As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equality for some transformationTof the form (1.4), (1.5). More generally for fixedk except on a set whose probability is small whenF_nandM_nare large. If the theorems of Section 3 or 4 apply, T^k(f_n(·),m_n(·)) will be close to a fixed vector ζ whenkis large and thus there is hope thatf_n(·) andm_n(·) will converge, onceF_nandM_nbecome large. We therefore have to put on some conditions which will makeF_nandM_ngrow. 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 thenth generation, may actually be smaller than the size of thenth generation for certain directionsf_n(·),m(·).