Inequalities for branching processes

Abstract

Summary: IfF(s) is the probability generating function of a non-negative random variable, thenthfunctional iterateF_n(s) =F_n–1(F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involvingF_n(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim_n→∞m⁻ⁿ{1−F_n(s)}, 0 ≦s≦ 1 wherem=F^′(1) < 1 andF^′′(1) < ∞; for the expected time to extinctionand for the limiting conditional-distribution generating function lim_n→∞{F_n(s) −F_n(0)} [1 –F_n(0)]^–1. Particular attention is paid to the caseF(s) = exp {m(s− 1)}.