Analysis of transient effects for branching processes

We study the limiting behavior of conditional distributions of nonhomogeneous Markov branching processes as time increases, when the mean numbers of direct offsprings of a particle are close to one uniformly over generations. The three following basic aspects are analyzed: stability of transient effects when the time homogeneity is violated, convergence rate in the integral limit theorem, and asymptotic expansions. We consider the discrete time processes only, but the results obtained can be also extended in a natural way (by embedding) to the continuous time case.