Theory of Probability and Mathematical Statistics
A new criterion for recurrence of Markov chains with an infinitely countable set of states
Vyacheslav M. Abramov
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Abstract: For a class of irreducible Markov chains with an infinitely countable set of states, we establish a new verifiable necessary and sufficient condition for recurrence. We show that if one of the basic assumptions is not satisfied, then the statement of the theorem becomes invalid.
Keywords: Discrete-time Markov chains, recurrence and transience, birth-and-death processes, continuous-time Markov processes, stochastic calculus
Bibliography: Vyacheslav M. Abramov, Extension of the Bertrand–De Morgan test and its application, Amer. Math. Monthly 127 (2020), no. 5, 444–448. MR 4095689, DOI 10.1080/00029890.2020.1722551
Vyacheslav M. Abramov, Necessary and sufficient conditions for the convergence of positive series, J. Class. Anal. 19 (2022), no. 2, 117–125. MR 4428668, DOI 10.7153/jca-2022-19-09
Vyacheslav M. Abramov, Conditions for recurrence and transience for time-inhomogeneous birth-and-death processes, Bull. Aust. Math. Soc. 109 (2024), no. 2, 393–402. MR 4715155, DOI 10.1017/S0004972723000539
Kai Lai Chung, Markov chains with stationary transition probabilities, Die Grundlehren der mathematischen Wissenschaften, Band 104, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 116388, DOI 10.1007/978-3-642-49686-8
Rick Durrett, Probability—theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 49, Cambridge University Press, Cambridge, 2019. Fifth edition of [ MR1068527]. MR 3930614, DOI 10.1017/9781108591034
Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. MR 1943877, DOI 10.1007/978-3-662-05265-5
Samuel Karlin and James McGregor, The classification of birth and death processes, Trans. Amer. Math. Soc. 86 (1957), 366–400. MR 94854, DOI 10.1090/S0002-9947-1957-0094854-8
Yaakov Kogan and Robert Sh. Liptser, Limit nonstationary behavior of large closed queueing networks with bottlenecks, Queueing Systems Theory Appl. 14 (1993), no. 1-2, 33–55. MR 1238661, DOI 10.1007/BF01153525
R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664, DOI 10.1007/978-94-009-2438-3
J. R. Norris, Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998. Reprint of 1997 original. MR 1600720
Philip E. Protter, Stochastic integration and differential equations, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR 2020294
