On a functional equation for general branching processes

If Z( t ) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W , then it is known that E ( W ) = 1 and that ϕ ( u ) = E ( e –uW ) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E ( W |log W | y ) < ∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E ( W 1 + β L ( W )) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.