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How many families survive for a long time
V. A. Vatutin, E. E. Dyakonova · 2016-08-29 · via math.PR updates on arXiv.org

A critical branching process $\left\{Z_{k},k=0,1,2,...\right\} $ in a random environment generated by a sequence of independent and identically distributed random reproduction laws is considered.\ Let $Z_{p,n}$ be the number of particles at time $p\leq n$ 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 $p\ll n$.

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