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Article

On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process

1
Department of Applied Mathematics, Vologda State University, IPI FRC CSC RAS, VolSC RAS, 160000 Vologda, Russia
2
Department of Mathematics, Vologda State University, 160000 Vologda, Russia
3
Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
4
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
5
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(5), 477; https://doi.org/10.3390/math7050477
Received: 15 April 2019 / Revised: 18 May 2019 / Accepted: 21 May 2019 / Published: 26 May 2019
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered. View Full-Text
Keywords: multidimensional birth-death process; inhomogeneous continuous-time Markov chain; rate of convergence; one dimensional projection multidimensional birth-death process; inhomogeneous continuous-time Markov chain; rate of convergence; one dimensional projection
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MDPI and ACS Style

Zeifman, A.; Satin, Y.; Kiseleva, K.; Korolev, V. On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process. Mathematics 2019, 7, 477. https://doi.org/10.3390/math7050477

AMA Style

Zeifman A, Satin Y, Kiseleva K, Korolev V. On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process. Mathematics. 2019; 7(5):477. https://doi.org/10.3390/math7050477

Chicago/Turabian Style

Zeifman, Alexander, Yacov Satin, Ksenia Kiseleva, and Victor Korolev. 2019. "On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process" Mathematics 7, no. 5: 477. https://doi.org/10.3390/math7050477

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