Abstract
We consider a critical homogeneous-continuous-time Markov branching system, i.e. the average value of the branching rate is zero. 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.
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A. Imomov conceived the relevance of the task and determined the way to solve it. M. Murtazaev, with the support of A. Imomov, implemented the idea of a method for solving the problem, and both wrote this article.
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Imomov, A.A., Murtazaev, M. (2023). Refined Limit Theorems for the Critical Continuous-Time Markov Branching Systems. In: Dudin, A., Nazarov, A., Moiseev, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2022. Communications in Computer and Information Science, vol 1803. Springer, Cham. https://doi.org/10.1007/978-3-031-32990-6_6
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