On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process
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 o...
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MDPI AG
2019-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/5/477 |
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| author | Alexander Zeifman Yacov Satin Ksenia Kiseleva Victor Korolev |
| author_facet | Alexander Zeifman Yacov Satin Ksenia Kiseleva Victor Korolev |
| author_sort | Alexander Zeifman |
| collection | DOAJ |
| description | 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. |
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| format | Article |
| id | doaj.art-a49a18a307fc4a88b5094db5dde11460 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-22T14:21:23Z |
| publishDate | 2019-05-01 |
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| series | Mathematics |
| spelling | doaj.art-a49a18a307fc4a88b5094db5dde114602022-12-21T18:22:58ZengMDPI AGMathematics2227-73902019-05-017547710.3390/math7050477math7050477On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death ProcessAlexander Zeifman0Yacov Satin1Ksenia Kiseleva2Victor Korolev3Department of Applied Mathematics, Vologda State University, IPI FRC CSC RAS, VolSC RAS, 160000 Vologda, RussiaDepartment of Mathematics, Vologda State University, 160000 Vologda, RussiaDepartment of Applied Mathematics, Vologda State University, 160000 Vologda, RussiaFaculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaWe 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.https://www.mdpi.com/2227-7390/7/5/477multidimensional birth-death processinhomogeneous continuous-time Markov chainrate of convergenceone dimensional projection |
| spellingShingle | Alexander Zeifman Yacov Satin Ksenia Kiseleva Victor Korolev On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process Mathematics multidimensional birth-death process inhomogeneous continuous-time Markov chain rate of convergence one dimensional projection |
| title | On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process |
| title_full | On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process |
| title_fullStr | On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process |
| title_full_unstemmed | On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process |
| title_short | On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process |
| title_sort | on the rate of convergence for a characteristic of multidimensional birth death process |
| topic | multidimensional birth-death process inhomogeneous continuous-time Markov chain rate of convergence one dimensional projection |
| url | https://www.mdpi.com/2227-7390/7/5/477 |
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