A Note on a Functional Equation Arising in Galton-Watson Branching Processes.
Abstract
The functional equation phi(mu) = h(phi(u)) where h(s) = the summation from j=o to infinity of p subj S supj is a p.g.f. with 1<m=h primed (1-) < infinity and phi(u) = the integral from o to infinity of e to the (-ut) power dF(t) where F(t) is a non-decreasing right continuous function with F(0-) = 0 and F(+ infinity) = 1 arises in Galton-Watson process in a natural way. One proves here that for any p>or= 0, the integral from o to infinity of t/logt/supp dF(t)< infinity if and only if the summation from j=2 to infinity of j((log j) sup(p+1))(p subj)< infinity. This unifies several results in the literature on supercritical Galton-Watson process. One generalizes this to an age dependent branching process case as well. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 1969
- Accession Number
- AD0715949
Entities
People
- Krishna B. Athreya
Organizations
- University of Wisconsin–Madison