Let ${Z_n}$ be a branching process whose offspring distributions
vary with n. It is shown that the sequence ${\max_{i>0} P(Z_n = i)}$
has a limit. Denote this limit by M. It turns out that M is
positive only if the offspring variables rapidly approach constants. Let
${c_n}$ be a sequence of constants and $W_n = Z_n / c_n$. It will be proven
that $M = 0$ is necessary and sufficient for the limit distribution functions
of all convergent ${W_n}$ to be continuous on $(0, \infty)$. If $M > 0$
there is, up to an equivalence, only one sequence ${c_n}$ such that ${W_n}$ has
a limit distribution with jump points in $(0, \infty)$. Necessary and
sufficient conditions for continuity of limit distributions are derived in
terms of the offspring distributions of ${Z_n}$.
@article{1034968232,
author = {Cohn, Harry},
title = {On the asymptotic patterns of supercritical branching processes in
varying environments},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 896-902},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968232}
}
Cohn, Harry. On the asymptotic patterns of supercritical branching processes in
varying environments. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 896-902. http://gdmltest.u-ga.fr/item/1034968232/