High level subcritical branching processes in a random environment

A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk Θ(t), t ≥ 0, is equal to 1 for some value of the argument ϰ > 0. Let Tx be the time when the process first attains the half-axis (x,+∞) and T be the lifetime of this process. It is shown that the random variable Tx/lnx, considered under the condition Tx < +∞, converges in distribution to a degenerate random variable equal to 1/Θ′(ϰ), and the random variable T/ ln x, considered under the same condition, converges in distribution to a degenerate random variable equal to 1/Θ′(ϰ) − 1/Θ′(0).