Generalized Markov branching processes with state-dependent offspring distributions

Abstract

The limiting behaviour of divergent time-homogeneous Markov processes on the non-negative integers which generalize the Markov branching processes is studied. The span of time between two jumps is exponentially distributed where the exponent depends linearly or sublinearly on the state. The offspring distributions are allowed to be state-dependent with expectations which are non-increasing while the state increases. Most of the results for ordinary supercritical Markov branching processes are generalized, including a necessary and sufficient condition for divergence (with natural rate), similar to the (x log x)-condition, and, in some special cases analoga of the CLT and LIL.