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
- A. V. Swishchuk, Semi-Markov random evolutions: some ideas, methods and results, Exploring Stochastic Laws. Festschrift in Honour of the 70th Birthday of Academician V. S. Korolyuk (A. V. Skorokhod and Yu. V. Borovskikh, eds.), VSP, 1995, pp. 417–442. MR 1714025
- 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
