The critical age-dependent branching process with offspring p.g.f. of the form $f(s) = s + (1 - s)^{1 + \alpha}L(1 - s), 0 < \alpha \leq 1, L$ slowly varying at 0, is investigated. We generalize Kesten's unpublished necessary condition to establish N.A.S.C. on the tail of the lifetime distribution for existence of a nondegenerate normalized conditioned limit law and pose several related questions.
@article{1176994421,
author = {Goldstein, Martin I. and Hoppe, Fred M.},
title = {Necessary and Sufficient Lifetime Conditions for Normed Convergence of Critical Age-Dependent Processes with Infinite Variance},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 490-497},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994421}
}
Goldstein, Martin I.; Hoppe, Fred M. Necessary and Sufficient Lifetime Conditions for Normed Convergence of Critical Age-Dependent Processes with Infinite Variance. Ann. Probab., Tome 9 (1981) no. 6, pp. 490-497. http://gdmltest.u-ga.fr/item/1176994421/