A bound for the remainder term in the asymptotic expansion of a functional constructed from a semi-Markov random evolution
Authors:
V. S. Koroliuk and I. V. Samoilenko
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 98 (2019), 217-227
MSC (2010):
Primary 60J25; Secondary 35C20
DOI:
https://doi.org/10.1090/tpms/1072
Published electronically:
August 19, 2019
MathSciNet review:
3824688
Full-text PDF
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Additional Information
Abstract: The regular as well as singular component of the asymptotic expansion of a functional constructed from a semi-Markov random evolution is found, and regularity of the initial data is shown in Theory Probab. Math. Statist. 96 (2018), 83–100. A bound for the remainder term of the asymptotic expansion obtained in the above-mentioned paper is studied in the current paper.
References
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- Vladimir S. Korolyuk and Vladimir V. Korolyuk, Stochastic models of systems, Mathematics and its Applications, vol. 469, Kluwer Academic Publishers, Dordrecht, 1999. MR 1753470
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- V. S. Korolyuk and Ī. V. Samoĭlenko, Asymptotic expansion of a functional of semi-Markov random evolution in a diffusion approximation scheme, Teor. Ĭmovīr. Mat. Stat. 96 (2017), 84–99 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 96 (2018), 83–100. MR 3666873, DOI https://doi.org/10.1090/tpms/1035
- A. A. Pogorui and Ramón M. Rodríguez-Dagnino, Asymptotic expansion for transport processes in semi-Markov media, Teor. Ĭmovīr. Mat. Stat. 83 (2010), 106–112; English transl., Theory Probab. Math. Statist. 83 (2011), 127–134. MR 2768853, DOI https://doi.org/10.1090/S0094-9000-2012-00846-6
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References
- S. Albeverio, V. S. Koroliuk, and I. V. Samoilenko, Asymptotic expansion of semi-Markov random evolutions, Stochastics 81 (2009), no. 5, 343–356. MR 2569263
- V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Kluwer Academic Publisher, Dordrecht, 1999. MR 1753470
- V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, New Jersey, 2005. MR 2205562
- V. S. Korolyuk and A. F. Turbin, Mathematical foundation of state lumping of large systems, Kluwer Academic Publisher, Dordrecht, 1990. MR 1281385
- V. S. Koroliuk and I. V. Samoilenko, Asymptotic expansion for a functional of semi-Markov random evolution in diffusion approximation scheme, Teor. Imovirnost. Matem. Statyst. 96 (2017), 84–99; English transl. in Theory Probab. Math. Statist. 96 (2018), 83–100. MR 3666873
- A. A. Pogorui and R. M. Rodriguez-Dagnino, Asymptotic expansion for transport processes in semi-Markov media, Teor. Imovirnost. Matem. Statyst. 83 (2010), 106–112; English transl. in Theory Probab. Math. Statist. 83 (2011), 127–134. MR 2768853
- I. V. Samoilenko, Asymptotic expansion for the functional of Markovian evolution in $\mathbb {R}^d$ in the circuit of diffusion approximation, J. Appl. Math. Stoch. Anal. 3 (2005), 247–258. MR 2203032
- I. V. Samoilenko, Asymptotic expansion of a semi-Markov random evolution, Ukrain. Matem. Zh. 58 (2006), no. 9, 1396–1414; English transl. Ukr. Math. J. 58 (2006), no. 9, 1234-1248. MR 2345090
- V. M. Shurenkov, Ergodic Markov processes, “Nauka”, Moscow, 1989. (Russian) MR 1087782
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- A. Swishchuk and Jianhong Wu, Evolution of Biological Systems in Random Media: Limit Theorems and Stability, Springer, New York, 2003. MR 2031169
- A. V. Swishchuk, Estimations of convergence rate in limit theorems for semi-Markov random evolutions, Stochastic Systems and Their Applications, Inst. of Math. Ukrainian SSR, Kiev, 1990, pp. 86–92. (Russian) MR 1095107
- G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: a Singular Perturbation Approach, Springer, New York, 1998. MR 1488963
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Additional Information
V. S. Koroliuk
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv, Ukraine 01601
Email:
vskorol@yahoo.com
I. V. Samoilenko
Affiliation:
Faculty of Computer Science and Cybernetics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
isamoil@i.ua
Keywords:
Asymptotic expansion,
semi-Markov process,
random evolution,
remainder term
Received by editor(s):
February 16, 2018
Published electronically:
August 19, 2019
Additional Notes:
This paper as well as its first part \cite{KoSam} contains results obtained under the support given by the Presidential Grant 0117U007015 from the State Fund of Fundamental Researches of Ukraine
Article copyright:
© Copyright 2019
American Mathematical Society