How many families survive for a long time?

Let $\{ Z_{k},k=0,1,\ldots\}$ be a critical branching process in a random environment generated by a sequence of independent and identically distributed random reproduction laws, and let $Z_{p,n}$ be the number of particles at time $p\le n$ having a positive offspring number at time $n$. A theorem is proved describing the limiting behavior, as $n\rightarrow \infty$, of the distribution of a properly scaled process $\log Z_{p,n}$ under the assumptions $Z_{n}>0$ and $p\ll n$.