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A large-deviation principle for birth–death processes with a linear rate of downward jumps

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  31 October 2023

Artem Logachov ,
Yuri Suhov ,
Nikita Vvedenskaya  and
Anatoly Yambartsev
Artem Logachov*
Affiliation:
Novosibirsk State University; Novosibirsk State Technical University; Sobolev Institute of Mathematics
Yuri Suhov*
Affiliation:
Penn State University; University of Cambridge, St John’s College, Cambridge
Nikita Vvedenskaya*
Affiliation:
Institute for Information Transmission Problems
Anatoly Yambartsev*
Affiliation:
University of São Paulo
*
*Postal address: Novosibirsk State University, 2, Pirogova str., Novosibirsk, 630090, Russia. Novosibirsk State Technical University, pr. K. Marksa, 20, Novosibirsk, 630073, Russia. Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia. Email: omboldovskaya@mail.ru
**Postal address: Department of Mathematics, Penn State University, PA 16802, USA. Email: ims14@psu.edu
***Postal address: Kharkevich Institute for Information Transmission Problems RAS, Bol'shoy Karetnyi per., 19, Moscow, 127051, Russia. Email: ndv@iitp.ru
****Postal address: University of São Paulo Institute of Mathematics and Statistics, 05508-090, Rua do Matao, 1010, São Paulo, SP, Brazil. Email: yambar@usp.br
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Abstract

Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

Keywords

Large-deviation principlelocal large-deviation principlebirth–death processesrate functional

MSC classification

Primary: 60F10: Large deviations
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 3 , September 2024 , pp. 781 - 801
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Borovkov, A. A. and Mogulskii, A. A. (2012). Large deviation principles for random walk trajectories. I. Theory Prob. Appl. 56, 538–561.10.1137/S0040585X97985613CrossRefGoogle Scholar
Borovkov, A. A. and Mogulskii, A. A. (2013). Large deviation principles for random walk trajectories. II. Theory Prob. Appl. 57, 1–27.10.1137/S0040585X97985765CrossRefGoogle Scholar
Borovkov, A. A. and Mogulskii, A. A. (2013). Inequalities and principles of large deviations for the trajectories of processes with independent increments. Siberian Math. J. 54, 217–226.10.1134/S0037446613020055CrossRefGoogle Scholar
Borovkov, A. A. and Mogulskii, A. A. (2014). Large deviation principles for random walk trajectories. III. Theory Prob. Appl. 58, 25–37.10.1137/S0040585X97986370CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (2009). Large Deviations Techniques and Applications. Springer, Berlin.Google Scholar
Deutschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, New York.Google Scholar
Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes (Math. Surveys Monographs 131). American Mathematical Society, Providence, RI.10.1090/surv/131/04CrossRefGoogle Scholar
Gnedenko, B. V., Belyayev, Yu. K. and Solovyev, A. D. (1969). Mathematical Methods of Reliability Theory. Academic Press, New York.Google Scholar
Karlin, S. and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.10.1090/S0002-9947-1957-0094854-8CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1958). Linear growth, birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Kelbert, M., Stuhl, I. and Suhov, Y. (2018). Weighted entropy and optimal portfolios for risk-averse Kelly investments. Aequat. Math. 92, 165–200.10.1007/s00010-017-0515-6CrossRefGoogle Scholar
Kendall, D. G. (1960). Birth-and-death processes, and the theory of carcinogenesis. Biometrika 47, 13–21.10.1093/biomet/47.1-2.13CrossRefGoogle Scholar
Logachov, A. V., Suhov, Y. M., Vvedenskaya, N. D. and Yambartsev, A. A. (2020). A remark on normalizations in a local large deviations principle for an inhomogeneous birth-and-death process. Siberian Electron. Math. Rep. 17, 1258–1269.Google Scholar
Lynch, I. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Prob. 15, 610–627.10.1214/aop/1176992161CrossRefGoogle Scholar
Mazel, A., Suhov, Y., Stuhl, I. and Zohren, S. (2014). Dominance of most tolerant species in multi-type lattice Widom–Rowlinson models. J. Statist. Mech. 2014, P08010.10.1088/1742-5468/2014/8/P08010CrossRefGoogle Scholar
Mogulskii, A. A. (2017) The large deviation principle for a compound Poisson process. Siberian Adv. Math. 27, 160–186.10.3103/S1055134417030026CrossRefGoogle Scholar
Mogulskii, A., Pechersky, E. and Yambartsev, A. (2014). Large deviations for excursions of nonhomogeneous Markov processes. Electron. Commun. Prob. 19, 1–8.10.1214/ECP.v19-3289CrossRefGoogle Scholar
Natanson, I. P. (2016). Theory of Functions of a Real Variable. Dover, New York.Google Scholar
Novozhilov, A. S., Karev, G. P. and Koonin, E. V. (2006). Biological applications of the theory of birth-and-death processes. Briefings Bioinform. 7, 70–85.10.1093/bib/bbk006CrossRefGoogle ScholarPubMed
Puhalskii, A. A. (1991) On functional principle of large deviations. In New Trends in Probability and Statistics, eds. V. V. Sazonov and T. L. Shervashidze. VSP/Mokslas, Utrecht, pp. 198–219.Google Scholar
Riesz, F. and Szökefalvi-Nagy, B. (1990). Functional Analysis. Dover, New York.Google Scholar
Robert, P. (2003). Stochastic Networks and Queues. Springer, Berlin.10.1007/978-3-662-13052-0CrossRefGoogle Scholar
Shi, Y., Jiang, X. and Ye, K. (2011). An energy-efficient scheme for cloud resource provisioning based on CloudSim. In Proc. IEEE Int. Conf. Cluster Computing, pp. 595–599.10.1109/CLUSTER.2011.63CrossRefGoogle Scholar
Stollenwerk, N. and Jansen, V. (2011). Population Biology and Criticality: From Critical Birth–Death Processes to Self-Organized Criticality in Mutation Pathogen Systems. World Scientific, Singapore.Google Scholar
Stroock, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York.10.1007/978-1-4613-8514-1CrossRefGoogle Scholar
Stuhl, I. and Suhov, Y. (2015). Selected data compression: A refinement of Shannon’s principle. In Analytical and Computational Methods in Probability Theory (Lect. Notes Comp. Sci. 10684). Springer, Berlin, pp. 309–321.Google Scholar
Stuhl, I. and Suhov, Y. (2016). Talk at the 7th European Congress of Mathematicians, Berlin.Google Scholar
Suhov, Y. and Stuhl, I. (2015). On principles of large deviation and selected data compression. Preprint, arXiv:1604.06971v1.Google Scholar
Valent, G. (1996). Exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials. J. Comp. Appl. Math. 67, 103–127.10.1016/0377-0427(94)00124-3CrossRefGoogle Scholar
Van Doorn, E. A. (2003). Birth–death processes and associated polynomials. J. Comp. Appl. Math. 153, 497–506.10.1016/S0377-0427(02)00594-0CrossRefGoogle Scholar
Varadhan, S. R. S. (1984). Large Deviations and Applications. SIAM, Philadelphia.10.1137/1.9781611970241CrossRefGoogle Scholar
Varadhan, S. R. S. (2008), Large deviations. Ann. Prob. 36, 397–419.10.1214/07-AOP348CrossRefGoogle Scholar
Vvedenskaya, N. D., Logachov, A. V., Suhov, Y. M. and Yambartsev, A. A. (2018). A local large deviation principle for inhomogeneous birth-and-death processes. Problems Inf. Trans. 54, 263–280.10.1134/S0032946018030067CrossRefGoogle Scholar
Vvedenskaya, N., Suhov, Y. and Belitsky, V. (2013). A non-linear model of trading mechanism on a financial market. Markov Proc. Relat. Fields 19, 83–98.Google Scholar