Accession Number : AD0715949

Title :   A Note on a Functional Equation Arising in Galton-Watson Branching Processes.

Descriptive Note : Technical summary rept.,

Corporate Author : WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER

Personal Author(s) : Athreya,Krishna B.

Report Date : JUL 1969

Pagination or Media Count : 17

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)

Descriptors :   (*STATISTICAL PROCESSES, DISTRIBUTION FUNCTIONS), (*STOCHASTIC PROCESSES, FUNCTIONAL ANALYSIS), RANDOM VARIABLES, ITERATIONS, SET THEORY, PROBABILITY, THEOREMS

Subject Categories : Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE