On Population-Size-Dependent Branching Processes
Advances in Applied Probability, 16 (1):
pp. 30-55 (1984)Abstract
We consider a stochastic model for the development in time of a population Zn where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance σ k 2 of offspring distribution stabilize as the population size k grows to ∞ , mk→ m, σ k 2→ σ 2. The process exhibits different asymptotic behaviour according to $m<1$ , m=1, $m>1$ ; moreover, the rate of convergence of mk to m plays an important role. It is shown that if $m<1$ or m=1 and mn approaches 1 not slower than n-2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n-1, then Zn/n converges in distribution to a gamma distribution, moreover $Z_n^-1=ınfty,\ Z_n^-2<$ a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches $m>1$ not slower than n-α, $>0$ , and σ n 2 do not grow to ∞ faster than nβ, $<1$ then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.