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2016 | OriginalPaper | Buchkapitel

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

verfasst von : Dmitrii Silvestrov, Sergei Silvestrov

Erschienen in: Engineering Mathematics II

Verlag: Springer International Publishing

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Abstract

New algorithms for computing asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.

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Zurück zum Zitat Kaplan, E.I.: Limit Theorems for Sum of Switching Random Variables with an Arbitrary Phase Space of Switching Component. Candidate of Science dissertation, Kiev State University (1980) Kaplan, E.I.: Limit Theorems for Sum of Switching Random Variables with an Arbitrary Phase Space of Switching Component. Candidate of Science dissertation, Kiev State University (1980)
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Zurück zum Zitat Kartashov, M.V.: Strong Stable Markov Chains, 138 pp. VSP, Utrecht and TBiMC, Kiev (1996) Kartashov, M.V.: Strong Stable Markov Chains, 138 pp. VSP, Utrecht and TBiMC, Kiev (1996)
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Zurück zum Zitat Kartashov, M.V.: Calculation of the spectral ergodicity exponent for the birth and death process. Ukr. Mat. Zh. 52, 889–897 (2000) (English translation in Ukr. Math. J. 52, 1018–1028) Kartashov, M.V.: Calculation of the spectral ergodicity exponent for the birth and death process. Ukr. Mat. Zh. 52, 889–897 (2000) (English translation in Ukr. Math. J. 52, 1018–1028)
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Zurück zum Zitat Khasminskii, R.Z., Yin, G., Zhang, Q.: Singularly perturbed Markov chains: quasi-stationary distribution and asymptotic expansion. In: Proceedings of Dynamic Systems and Applications, vol. 2, pp. 301–308. Atlanta, GA, 1995. Dynamic, Atlanta, GA (1996) Khasminskii, R.Z., Yin, G., Zhang, Q.: Singularly perturbed Markov chains: quasi-stationary distribution and asymptotic expansion. In: Proceedings of Dynamic Systems and Applications, vol. 2, pp. 301–308. Atlanta, GA, 1995. Dynamic, Atlanta, GA (1996)
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Zurück zum Zitat Khasminskii, R.Z., Yin, G., Zhang, Q.: Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math. 56(1), 277–293 (1996)MathSciNetMATHCrossRef Khasminskii, R.Z., Yin, G., Zhang, Q.: Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math. 56(1), 277–293 (1996)MathSciNetMATHCrossRef
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Zurück zum Zitat Kim, D.S., Smith, R.L.: An exact aggregation/disaggregation algorithm for large scale Markov chains. Nav. Res. Logist. 42(7), 1115–1128 (1995)MathSciNetMATHCrossRef Kim, D.S., Smith, R.L.: An exact aggregation/disaggregation algorithm for large scale Markov chains. Nav. Res. Logist. 42(7), 1115–1128 (1995)MathSciNetMATHCrossRef
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Zurück zum Zitat Konstantinov, M. Gu, D.W., Mehrmann, V., Petkov, P.: Perturbation Theory for Matrix Equations. Studies in Computational Mathematics, vol. 9, xii+429 pp. North-Holland, Amsterdam (2003) Konstantinov, M. Gu, D.W., Mehrmann, V., Petkov, P.: Perturbation Theory for Matrix Equations. Studies in Computational Mathematics, vol. 9, xii+429 pp. North-Holland, Amsterdam (2003)
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Zurück zum Zitat Konstantinov, M.M., Petkov, P.H.: Perturbation methods in linear algebra and control. Appl. Comput. Math. 7(2), 141–161 (2008)MathSciNetMATH Konstantinov, M.M., Petkov, P.H.: Perturbation methods in linear algebra and control. Appl. Comput. Math. 7(2), 141–161 (2008)MathSciNetMATH
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Zurück zum Zitat Kontoyiannis, I., Meyn, S.P.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13(1), 304–362 (2003)MathSciNetMATHCrossRef Kontoyiannis, I., Meyn, S.P.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13(1), 304–362 (2003)MathSciNetMATHCrossRef
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Zurück zum Zitat Korolyuk, D.V.: Limit theorems for hitting time type functionals defined on processes with semi-Markov switchings. Candidate of Science Dissertation, Kiev State University (1983) Korolyuk, D.V.: Limit theorems for hitting time type functionals defined on processes with semi-Markov switchings. Candidate of Science Dissertation, Kiev State University (1983)
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Zurück zum Zitat Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding domains for ergodic Markov chains. Teor. Veroyatn. Primen. 28, 410–420 (1983). (English translation in Theory Probab. Appl. 28, 432–442) Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding domains for ergodic Markov chains. Teor. Veroyatn. Primen. 28, 410–420 (1983). (English translation in Theory Probab. Appl. 28, 432–442)
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Zurück zum Zitat Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding regions for processes with semi-Markov switchings. Teor. Veroyatn. Primen. 29, 539–544 (1984). (English translation in Theory Probab. Appl. 29, 558–563) Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding regions for processes with semi-Markov switchings. Teor. Veroyatn. Primen. 29, 539–544 (1984). (English translation in Theory Probab. Appl. 29, 558–563)
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Zurück zum Zitat Korolyuk, V.S.: On asymptotical estimate for time of a semi-Markov process being in the set of states. Ukr. Mat. Zh. 21, 842–845 (1969). (English translation in Ukr. Math. J. 21, 705–710) Korolyuk, V.S.: On asymptotical estimate for time of a semi-Markov process being in the set of states. Ukr. Mat. Zh. 21, 842–845 (1969). (English translation in Ukr. Math. J. 21, 705–710)
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Zurück zum Zitat Korolyuk, V.S.: Stochastic Models of Systems, 208 pp. Naukova Dumka, Kiev (1989) Korolyuk, V.S.: Stochastic Models of Systems, 208 pp. Naukova Dumka, Kiev (1989)
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Zurück zum Zitat Korolyuk, V.S., Brodi, S.M., Turbin, A.F.: Semi-Markov processes and their application. Probability Theory. Mathematical Statistics. Theoretical Cybernetics, vol. 11, pp. 47–97. VINTI, Moscow (1974) Korolyuk, V.S., Brodi, S.M., Turbin, A.F.: Semi-Markov processes and their application. Probability Theory. Mathematical Statistics. Theoretical Cybernetics, vol. 11, pp. 47–97. VINTI, Moscow (1974)
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Zurück zum Zitat Korolyuk, V.S., Korolyuk, V.V.: Stochastic Models of Systems. Mathematics and its Applications, vol. 469, xii+185 pp. Kluwer, Dordrecht (1999) Korolyuk, V.S., Korolyuk, V.V.: Stochastic Models of Systems. Mathematics and its Applications, vol. 469, xii+185 pp. Kluwer, Dordrecht (1999)
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Diffusion approximation of integral functionals in merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 59, 99–105 (1998). (English translation in Theory Probab. Math. Stat. 59, 101–107) Korolyuk, V.S., Limnios, N.: Diffusion approximation of integral functionals in merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 59, 99–105 (1998). (English translation in Theory Probab. Math. Stat. 59, 101–107)
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Diffusion approximation for integral functionals in the double merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 60, 77–84 (1999). (English translation in Theory Probab. Math. Stat. 60, 87–94) Korolyuk, V.S., Limnios, N.: Diffusion approximation for integral functionals in the double merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 60, 77–84 (1999). (English translation in Theory Probab. Math. Stat. 60, 87–94)
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Evolutionary systems in an asymptotic split phase space. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 145–161. Birkhäuser, Boston (2000)CrossRef Korolyuk, V.S., Limnios, N.: Evolutionary systems in an asymptotic split phase space. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 145–161. Birkhäuser, Boston (2000)CrossRef
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Markov additive processes in a phase merging scheme. In: Korolyuk, V., Prokhorov, Yu., Khokhlov, V., Klesov, O. (eds.) Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko, Kiev (2002). (Theory Stoch. Process., 8(3–4), 213–225) Korolyuk, V.S., Limnios, N.: Markov additive processes in a phase merging scheme. In: Korolyuk, V., Prokhorov, Yu., Khokhlov, V., Klesov, O. (eds.) Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko, Kiev (2002). (Theory Stoch. Process., 8(3–4), 213–225)
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space. Ann. Appl. Probab. 14, 489–516 (2004)MathSciNetMATHCrossRef Korolyuk, V.S., Limnios, N.: Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space. Ann. Appl. Probab. 14, 489–516 (2004)MathSciNetMATHCrossRef
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Zurück zum Zitat Korolyuk, V.S., Limnios, N.: Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 70, 63–73 (2004). (English translation in Theory Probab. Math. Stat. 70, 71–82) Korolyuk, V.S., Limnios, N.: Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 70, 63–73 (2004). (English translation in Theory Probab. Math. Stat. 70, 71–82)
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Zurück zum Zitat Koroliuk, V.S., Limnios, N.: Stochastic Systems in Merging Phase Space, xv+331 pp. World Scientific, Singapore (2005) Koroliuk, V.S., Limnios, N.: Stochastic Systems in Merging Phase Space, xv+331 pp. World Scientific, Singapore (2005)
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Zurück zum Zitat Koroliuk, V.S., Limnios, N.: Reliability of semi-Markov systems with asymptotic merging phase space. In: Rykov, V.V., Balakrishnan, N., Nikulin, M.S. (eds.) Mathematical and Statistical Models and Methods in Reliability, pp. 3–18. Birkhäuser, Boston (2010)CrossRef Koroliuk, V.S., Limnios, N.: Reliability of semi-Markov systems with asymptotic merging phase space. In: Rykov, V.V., Balakrishnan, N., Nikulin, M.S. (eds.) Mathematical and Statistical Models and Methods in Reliability, pp. 3–18. Birkhäuser, Boston (2010)CrossRef
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Zurück zum Zitat Korolyuk, V.S., Penev, I.P., Turbin, A.F.: The asymptotic behavior of the distribution of the absorption time of a Markov chain. Kibernetika (2), 20–22 (1972) Korolyuk, V.S., Penev, I.P., Turbin, A.F.: The asymptotic behavior of the distribution of the absorption time of a Markov chain. Kibernetika (2), 20–22 (1972)
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Zurück zum Zitat Korolyuk, V.S., Penev, I.P., Turbin, A.F.: Asymptotic expansion for the distribution of the absorption time of a weakly inhomogeneous Markov chain. In: Korolyuk, V.S. (ed.) Analytic Methods of Investigation in Probability Theory, pp. 97–105. Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1981) Korolyuk, V.S., Penev, I.P., Turbin, A.F.: Asymptotic expansion for the distribution of the absorption time of a weakly inhomogeneous Markov chain. In: Korolyuk, V.S. (ed.) Analytic Methods of Investigation in Probability Theory, pp. 97–105. Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1981)
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Zurück zum Zitat Korolyuk, V., Swishchuk, A.: Semi-Markov Random Evolutions, 254 pp. Naukova Dumka, Kiev (1992). (English revised edition of Semi-Markov Random Evolutions. Mathematics and its Applications, vol. 308, x+310 pp. Kluwer, Dordrecht, 1995) Korolyuk, V., Swishchuk, A.: Semi-Markov Random Evolutions, 254 pp. Naukova Dumka, Kiev (1992). (English revised edition of Semi-Markov Random Evolutions. Mathematics and its Applications, vol. 308, x+310 pp. Kluwer, Dordrecht, 1995)
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Zurück zum Zitat Korolyuk, V.V., Tadzhiev, A.: Asymptotic behavior of Markov evolutions prior to the time of absorption. Ukr. Mat. Zh. 38, 248–251 (1986). (English translation in Ukr. Math. J. 38, 219–222) Korolyuk, V.V., Tadzhiev, A.: Asymptotic behavior of Markov evolutions prior to the time of absorption. Ukr. Mat. Zh. 38, 248–251 (1986). (English translation in Ukr. Math. J. 38, 219–222)
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Zurück zum Zitat Korolyuk, V.S., Turbin, A.F.: On the asymptotic behaviour of the occupation time of a semi-Markov process in a reducible subset of states. Teor. Veroyatn. Mat. Stat. 2, 133–143 (1970). (English translation in Theory Probab. Math. Stat. 2, 133–143) Korolyuk, V.S., Turbin, A.F.: On the asymptotic behaviour of the occupation time of a semi-Markov process in a reducible subset of states. Teor. Veroyatn. Mat. Stat. 2, 133–143 (1970). (English translation in Theory Probab. Math. Stat. 2, 133–143)
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Zurück zum Zitat Korolyuk, V.S., Turbin, A.F.: A certain method of proving limit theorems for certain functionals of semi-Markov processes. Ukr. Mat. Zh. 24, 234–240 (1972)MathSciNetMATH Korolyuk, V.S., Turbin, A.F.: A certain method of proving limit theorems for certain functionals of semi-Markov processes. Ukr. Mat. Zh. 24, 234–240 (1972)MathSciNetMATH
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Zurück zum Zitat Korolyuk, V.S., Turbin, A.F.: Semi-Markov Processes and its Applications, 184 pp. Naukova Dumka, Kiev (1976) Korolyuk, V.S., Turbin, A.F.: Semi-Markov Processes and its Applications, 184 pp. Naukova Dumka, Kiev (1976)
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Zurück zum Zitat Silvestrov, D.S.: Nonlinearly perturbed Markov chains and large deviations for lifetime functionals. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 135–144. Birkhäuser, Boston (2000)CrossRef Silvestrov, D.S.: Nonlinearly perturbed Markov chains and large deviations for lifetime functionals. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 135–144. Birkhäuser, Boston (2000)CrossRef
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Zurück zum Zitat Silvestrov, D.S., Abadov, Z.A.: Asymptotic behaviour for exponential moments of sums of random variables defined on exponentially ergodic Markov chains. Dokl. Acad. Nauk Ukr. SSR, Ser. A (4), 23–25 (1984) Silvestrov, D.S., Abadov, Z.A.: Asymptotic behaviour for exponential moments of sums of random variables defined on exponentially ergodic Markov chains. Dokl. Acad. Nauk Ukr. SSR, Ser. A (4), 23–25 (1984)
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Zurück zum Zitat Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. Theory Stoch. Process., 12(28), no. 3–4, Part I: 151–186. Part II, 187–202 (2006)MATH Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. Theory Stoch. Process., 12(28), no. 3–4, Part I: 151–186. Part II, 187–202 (2006)MATH
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Zurück zum Zitat Turbin, A.F.: On asymptotic behavior of time of a semi-Markov process being in a reducible set of states. Linear case. Teor. Veroyatn. Mat. Stat. 4, 179–194 (1971). (English translation in Theory Probab. Math. Stat. 4, 167–182) Turbin, A.F.: On asymptotic behavior of time of a semi-Markov process being in a reducible set of states. Linear case. Teor. Veroyatn. Mat. Stat. 4, 179–194 (1971). (English translation in Theory Probab. Math. Stat. 4, 167–182)
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Metadaten
Titel
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
verfasst von
Dmitrii Silvestrov
Sergei Silvestrov
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-42105-6_10