Harmonic continuous-time branching moments

Details Harmonic continuous-time branching moments Harmonic continuous-time branching moments

2005-11-02

arxiv.org/abs/math/0511058v2

Annals of Applied Probability 2006, Vol. 16, No. 4, 2078-2097

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000493 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000493

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 $m_1\ge1$. If, furthermore, $k$ is greater than a second threshold $m_2\ge m_1$, the normalized mean inverse population is at most $1/(k-m_2)$. We express $m_1$ and $m_2$ 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.

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