ИСТИНА

Let us consider a subcritical branching process in random environment. For a fairly long time researchers studied this process under the nonextinction condition and the three different types of its asymptotical behavior were discovered. We studied this process under the condition of attaining of the high level x supposing that the moment-generating function θ(t) of a step of the associated random walk is equal to 1 for some t=ϰ>0. At first an asymptotical formula for the probability of attaining of the level x→∞ was discovered. We proved two laws of large numbers: 1) for the time of attaining of the high level x and 2) for the lifetime of the process under consideration. It turns out that in the logarithmic scale the trend of trajectories of the process has an "up and down" form: the first straight-line begins at the point (0,0) with the slope θ′(ϰ)>0; the second straight-line begins at the point ((θ′(ϰ))⁻¹ln x,ln x) with the slope θ′(0)<0. We also proved a functional limit theorem which describes deviations of trajectories of the process from the trend. Тhе limit process is a standard Brownian motion.