Inequalities for branching processes
- 1 June 1966
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 3 (1) , 261-267
- https://doi.org/10.2307/3212050
Abstract
Summary: IfF(s) is the probability generating function of a non-negative random variable, thenthfunctional iterateFn(s) =Fn–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 involvingFn(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates limn→∞m−n{1−Fn(s)}, 0 ≦s≦ 1 wherem=F′(1) < 1 andF′′(1) < ∞; for the expected time to extinctionand for the limiting conditional-distribution generating function limn→∞{Fn(s) −Fn(0)} [1 –Fn(0)]–1. Particular attention is paid to the caseF(s) = exp {m(s− 1)}.Keywords
This publication has 2 references indexed in Scilit:
- The Theory of Branching ProcessesPublished by Springer Nature ,1963
- Reelle analytische Lösungen der Gleichung ϑ(ϑ(x))=ex und verwandter Funktionalgleichungen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1950