Harmonic continuous-time branching moments

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.