Springer Nature 2021 L A T E X template Monotone Ratio theorem convergence rate and its consequences in the theory of continuous-time Markov branching systems

Abstract We consider a critical homogeneous-continuous-time Markov branching system , i.e. the average value of the branching rate is one. Our basic assumption is that the branching rate generating function of the system regularly varies, in which slowly varying factor varies at infinity with an explicit expression remainder. We essentially rely on the improved version of the Basic Lemma of the critical Markov branching systems theory. First we establish a convergence rate in the Monotone ratio theorem. Subsequently we prove a local-convergence limit theorem on the asymptotic expansion of transition probabilities and their convergence to the invariant measure.MSC Classification: 60J80 , 60J85

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