Revisiting recurrence criteria of birth and death processes. Short proofs
Author:
Oleg K. Zakusylo
Journal:
Theor. Probability and Math. Statist. 107 (2022), 185-191
MSC (2020):
Primary 60J80; Secondary 60K25
DOI:
https://doi.org/10.1090/tpms/1182
Published electronically:
November 8, 2022
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Abstract: The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let ${T^\infty }$ be the passage time from zero to $\infty$. The regularity criterion says that ${T^\infty } < \infty$ if and only if $\mathbb {E}{T^\infty } < \infty$. It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.
References
- K.-L. Chung, Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967).
- Forrest W. Crawford, Lam Si Tung Ho, and Marc A. Suchard, Computational methods for birth-death processes, Wiley Interdiscip. Rev. Comput. Stat. 10 (2018), no. 2, e1423, 22. MR 3771529, DOI 10.1002/wics.1423
- Evgenii B. Dynkin and Aleksandr A. Yushkevich, Markov processes: Theorems and problems, Plenum Press, New York, 1969. Translated from the Russian by James S. Wood. MR 0242252, DOI 10.1007/978-1-4899-5591-3
- Ī. Ī. Ēžov and V. M. Šurenkov, Ergodic theorems connected with the Markov property of random processes, Teor. Verojatnost. i Primenen. 21 (1976), no. 3, 635–639 (Russian, with English summary). MR 0420844
- Yu Gong, Yong-Hua Mao, and Chi Zhang, Hitting time distributions for denumerable birth and death processes, J. Theoret. Probab. 25 (2012), no. 4, 950–980. MR 2993011, DOI 10.1007/s10959-012-0436-1
- 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
- S. Karlin and J. L. McGregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489–546. MR 91566, DOI 10.1090/S0002-9947-1957-0091566-1
- Samuel Karlin and James McGregor, Coincidence properties of birth and death processes, Pacific J. Math. 9 (1959), 1109–1140. MR 114247, DOI 10.2140/pjm.1959.9.1109
References
- K.-L. Chung, Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967).
- F. W. Crawford, L. S. T. Ho, and M. A. Suchard, Computational methods for birth-death processes, Wiley Interdiscip. Rev. Comput. Stat. 10 (2018), no. 2, e1423, 22. MR 3771529
- E. B. Dynkin and A. A. Yushkevich, Markov Processes: Theorems and Problems, Plenum Press, New York, 1969, Translated from the Russian by James S. Wood. MR 0242252
- I.I. Ezhov and V.M. Shurenkov, Ergodic theorems connected with the Markov property of random processes, Theory of Probability & Its Applications 21 (1976), no. 3, 635–639. MR 0420844
- Y. Gong, Y.-H. Mao, and C. Zhang, Hitting time distributions for denumerable birth and death processes, J. Theoret. Probab. 25 (2012), no. 4, 950–980. MR 2993011
- S. Karlin and J. McGregor, The classification of birth and death processes, Trans. Amer. Math. Soc. 86 (1957), 366–400. MR 94854
- S. Karlin and J. McGregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489–546. MR 91566
- S. Karlin and J. McGregor, Coincidence properties of birth and death processes, Pacific J. Math. 9 (1959), 1109–1140. MR 114247
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Additional Information
Oleg K. Zakusylo
Affiliation:
Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Email:
zakusylo@knu.ua
Keywords:
Classification of birth and death processes,
monotone convergence of birth and death processes
Received by editor(s):
February 6, 2022
Accepted for publication:
May 8, 2022
Published electronically:
November 8, 2022
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv
