Harmonic continuous-time branching moments

Abstract

We show that the mean inverse populations of nondecreasing, square integrable, continuous-time branching processes decrease to zero like the inverse of their mean population if and only if the initial population k is greater than a first threshold m₁≥1. If, furthermore, k is greater than a second threshold m₂≥m₁, the normalized mean inverse population is at most 1/(k−m₂). We express m₁ and m₂ as explicit functionals of the reproducing distribution, we discuss some analogues for discrete time branching processes and link these results to the behavior of random products involving i.i.d. nonnegative sums.