Functional equations and the Galton-Watson process
- 1 January 1969
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 1 (1) , 1-42
- https://doi.org/10.2307/1426407
Abstract
In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Letdenote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is thatfj≠ 1,j= 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 <f0≡F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)Keywords
This publication has 31 references indexed in Scilit:
- A LIMIT THEOREM FOR THE GALTON‐WATSON PROCESS WITH IMMIGRATIONAustralian Journal of Statistics, 1969
- On Recent Theorems Concerning the Supercritical Galton-Watson ProcessThe Annals of Mathematical Statistics, 1968
- On asymptotic properties of sub-critical branching processesJournal of the Australian Mathematical Society, 1968
- A branching process with mean one and possibly infinite varianceProbability Theory and Related Fields, 1968
- The Galton-Watson process with mean oneJournal of Applied Probability, 1967
- Note on Iteration of Concave FunctionsThe American Mathematical Monthly, 1967
- A Limit Theorem for Multidimensional Galton-Watson ProcessesThe Annals of Mathematical Statistics, 1966
- Spectral theory of branching processes. IProbability Theory and Related Fields, 1966
- Reelle analytische Lösungen der Gleichung ϑ(ϑ(x))=ex und verwandter Funktionalgleichungen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1950
- The probability distribution of gene-differences in relation to selection, mutation, and random extinctionMathematical Proceedings of the Cambridge Philosophical Society, 1949