Quadratic transformations: a model for population growth. II

Abstract

In this last part theFn(i) andMn(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 whenFnandMnare large,Fn + 1(i) andMn + 1(i) will, with high probability, be close to a certain function ofFn(·) andMn(·) (roughly the conditional expectation ofFn+1(i) andMn + 1(i) givenFn(·) andMn(·)). As we already indicated in Section 2, this leads (outside the exceptional set) to the approximate equalityfor some transformationTof the form (1.4), (1.5). More generally for fixedkexcept on a set whose probability is small whenFnandMnare large. If the theorems of Section 3 or 4 apply, Tk(fn(·),mn(·)) will be close to a fixed vector ζ whenkis large and thus there is hope thatfn(·) andmn(·) will converge, onceFnandMnbecome large. We therefore have to put on some conditions which will makeFnandMngrow. 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 directionsfn(·),m(·).