2005/11/30 by Didier Piau
Decision Sciences · Mathematics · #Markov Chains and Monte Carlo Methods #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60J80
paper · pdf · doi:10.1214/105051606000000493
published as Annals of Applied Probability 2006, Vol. 16, No. 4, 2078-2097 · Published at http://dx.doi.org/10.1214/105051606000000493 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/11/01 · arxiv created 2007/02/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.