Cohn, H. ``Branching processes in varying environment'' Statistics
Department Research Report #13, 1994.
Abstract
Let $\{Z_n\}$ be a branching process whose offspring distributions vary
with $n$, and write $M_n=\max_i P(Z_n=i)$. It is shown that the only
case when a limiting distribution function of a sequence $\{Z_n/c_n\}$,
for some constants $\{c_n\}$, has a jump away from $0$ is when
$\lim_{n \rightarrow \infty} M_n > 0$. A characterization of the
latter case in terms of offspring probabilities is given.