Cohn, H.: On the support and continuity of the limit distribution
of a branching process in varying environments.
Abstract
Let $\{Z_n\}$ be a branching process whose offspring distributions vary with
$n$, $\{c_n\}$ a sequence of constants and $W_n = Z_n / c_n$. Write $F$ for
the limit distribution of an a.s. convergent $\{W_n\}$. A necessary and
sufficient condition is given for the support of $F$ to be $[0,\infty]$.
Another necessary and sufficient condition is given for the continuity of
$F$ on $(0,\infty)$. Both conditions are formulated in terms of the
offspring variables of the process $\{X_n\}$. The support of $F$ depends
on the positivity or otherwise of $\{ P(X_n = i)\}$, while the continuity
requires the divergence of a series derived from $\{ P(X_n = i )\}$.
Examples of various types of limit distributions are given.