Abstract:
Suppose that in the same random environment, for each $n\in N$, a
subcritical branching process $Z^{\left( n\right) }=\left\{ Z_{i}^{\left(
n\right) },\text{ }i=0,1,\ldots \right\} $ is given such that $Z_{0}^{\left(
n\right) }=n$. A limit theorem is established for the extinction time $
T^{\left( n\right) }$ of the process $Z^{\left( n\right) }$, as $
n\rightarrow \infty $. In addition, functional limit theorems for two random
processes are proved: in the first of them, the number of particles of the
process $Z^{\left( n\right) }$ in the generation with number $\left\lfloor
t\ln n\right\rfloor $ is corresponded to the time $t$, and in the second,
the logarithm of this number of particles.
