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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References
Asmussen, S., Hering, H.: Branching processes. Birkhäuser, Boston (1983)
Athreya, K.B., Ney, P.E.: Branching processes. Springer, New York (1972). https://doi.org/10.1007/978-3-642-65371-1
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Univ. Press, Cambridge (1987)
Li, J., Cheng, L., Li, L.: Long time behaviour for Markovian branching-immigration systems. Discrete Event Dyn. Syst. 31(1), 37–57 (2021)
Feller, W.: An introduction to probability theory and its applications, v.2. JW & Sons (1971)
Harris, T.E.: The Theory of Branching Processes. Springer-Verlag, Berlin (1963)
Heathcote C. R. A branching process allowing immigration. Jour. Royal Stat. Soc. B-27(1), 138–143 (1965)
Imomov, A.A.: On a limit structure of the Galton-Watson branching processes with regularly varying generating functions. Prob. and math. stat. 39(1), 61–73 (2019)
Imomov, A.A., Meyliyev, A.: On application of slowly varying functions with remainder in the theory of Markov Branching Processes with mean one and infinite variance. Ukr. Math. J. 73(8), 1225–1237 (2022)
Imomov A.A., Tukhtaev E.E.: On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance. Stochastic Models, Publ. (2022). https://doi.org/10.1080/15326349.2022.2033628
Imomov A.A.: Limit properties of transition functions of continuous-time Markov branching processes. Int. J Stoch. Analysis 2014, 409345 (2014). https://doi.org/10.1155/2014/409345
Imomov, A.A.: On long-term behavior of continuous-time Markov branching processes allowing immigration. J Sib. Fed. Univ. Math. Phys. 7(4), 443–454 (2014)
Jagers, P.: Branching Progresses with Biological applications. Pitman Press, GB, JW & Sons (1975)
Pakes, A.G.: Critical Markov branching process limit theorems allowing infinite variance. Adv. Appl. Prob. 42, 460–488 (2010)
Pakes, A.G.: Revisiting conditional limit theorems for the mortal simple branching process. Bernoulli 5(6), 969–998 (1999)
Pakes, A.G.: Some results for non-supercritical Galton-Watson process with immigration. Math. Biosci. 24, 71–92 (1975)
Seneta, E.: Regularly Varying Functions. Springer, Berlin (1972). https://doi.org/10.1007/BFb0079658
Sevastyanov, B.A.: Branching Processes. Nauka, Moscow, Russia (1971)
Sevastyanov, B.A.: The theory of Branching stochastic process. Uspekhi Mathematicheskikh Nauk 6(46), 47–99 (1951)
Slack, R.S.: A branching process with mean one and possible infinite variance. Wahrscheinlichkeitstheor. und Verv. Geb. 9, 139–145 (1968)
Zolotarev, V.M.: More exact statements of several theorems in the theory of branching processes. Theory Prob. Appl. 2, 245–253 (1957)
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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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