Functional equations and the Galton-Watson process

Author:

Seneta E.

Abstract

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f0F(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.)

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics,Statistics and Probability

Reference74 articles.

1. Reuter G. E. H. (1968) (Private communication).

2. This ensures that the fn (x), x ∈ (0, d] are well defined for n ≧ 0, and in conjunction with e.g., the continuity of f(x) at 0, implies fn (x) ↓ 0 for fixed x.

3. In both results, remarks analogous to the footnote 6 to Theorem B concerning the finite intervals, 0 ≦ x ≦ d, 0 < x ≦ d (closed on the right) apply.

4. Papangelou F. (1967) A lemma on the Galton-Watson process. Z. Wahrscheinlichkeitsth. (to appear).

5. A branching process with mean one and possibly infinite variance

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