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

Part of: Limit theorems Markov processes

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
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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