Asymptotic behaviour of the survival probability of almost critical branching processes in a random environment (Q6569575)

Branching processes describe the development of a population of particles of a single type which, independently of one another, reproduce other particles of the same type. The simplest model of a branching process with discrete time is a so-called Galton-Watson process. Branching processes in random environment are a generalization of Galton-Watson branching processes. Unlike the Galton-Watson process, the distribution of the number of offspring in each generation in a branching process in a random environment depends on some random factor, called an environment. The critical branching process in a random environment \(\{ Z_k,\, k\geq 0\}\) and the triangular array scheme \(\{ Z_{k,n},\, 0\leq k\leq n,\, Z_{0,n}=1\}\) are considered. It is assumed that the \(Z_{k,n}\) are close to \(Z_k\) for large \(n\). Under certain assumptions it is established that\N\[\N\mathsf{P}(Z_{n,n}>0) \sim \mathsf{P}(Z_{n}>0) \sim \Upsilon\frac{e^{-c_-}}{\sqrt{\pi n}},\quad n\to\infty,\N\]\Nwhere \(\Upsilon\) is some positive constant and the constant \(c_-\) is determined through the associated random walk.