Necessary and sufficient conditions for convergence of first-rare-event-time processes for perturbed semi-Markov processes
Author:
D. S. Silvestrov
Journal:
Theor. Probability and Math. Statist. 95 (2017), 131-151
MSC (2010):
Primary 60J10, 60J22, 60J27, 60K15; Secondary 65C40
DOI:
https://doi.org/10.1090/tpms/1026
Published electronically:
February 28, 2018
MathSciNet review:
3631648
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Necessary and sufficient conditions for convergence in distribution of first-rare-event times and convergence in Skorokhod J-topology of first-rare-event-time processes for perturbed semi-Markov processes with finite phase space are obtained.
References
- V. V. Anisimov, Sluchaĭ nye protsessy s diskretnoĭ komponentoĭ, “Vishcha Shkola”, Kiev, 1988 (Russian). Predel′nye teoremy. [Limit theorems]. MR 955492
- Vladimir V. Anisimov, Switching processes in queueing models, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051
- Søren Asmussen, Applied probability and queues, 2nd ed., Applications of Mathematics (New York), vol. 51, Springer-Verlag, New York, 2003. Stochastic Modelling and Applied Probability. MR 1978607
- Søren Asmussen and Hansjörg Albrecher, Ruin probabilities, 2nd ed., Advanced Series on Statistical Science & Applied Probability, vol. 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. MR 2766220
- V. E. Bening and V. Yu. Korolev, Generalized Poisson Models and their Applications in Insurance and Finance, Modern Probability and Statistics, VSP, Utrecht, 2002.
- Myroslav Drozdenko, Weak convergence of first-rare-event times for semi-Markov processes. II, Theory Stoch. Process. 15 (2009), no. 2, 99–118. MR 2598530
- Boris V. Gnedenko and Victor Yu. Korolev, Random summation, CRC Press, Boca Raton, FL, 1996. Limit theorems and applications. MR 1387113
- Mats Gyllenberg and Dmitrii S. Silvestrov, Quasi-stationary phenomena in nonlinearly perturbed stochastic systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2456816
- Vladimir Kalashnikov, Geometric sums: bounds for rare events with applications, Mathematics and its Applications, vol. 413, Kluwer Academic Publishers Group, Dordrecht, 1997. Risk analysis, reliability, queueing. MR 1471479
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- Julian Keilson, Markov chain models—rarity and exponentiality, Applied Mathematical Sciences, vol. 28, Springer-Verlag, New York-Berlin, 1979. MR 528293
- D. V. Korolyuk and D. S. Sīl′vestrov, Time of reaching asymptotically receding domains for ergodic Markov chains, Teor. Veroyatnost. i Primenen. 28 (1983), no. 2, 410–420 (Russian, with English summary). MR 700223
- D. V. Korolyuk and D. S. Sil′vestrov, Attainment times of asymptotically receding domains for processes with semi-Markov switchings, Teor. Veroyatnost. i Primenen. 29 (1984), no. 3, 539–544 (Russian). MR 761139
- V. S. Koroljuk, The asymptotic behavior of the sojourn time of a semi-Markov process in a subset of the states, Ukrain. Mat. Ž. 21 (1969), 842–845 (Russian). MR 0251794
- Vladimir S. Korolyuk and Vladimir V. Korolyuk, Stochastic models of systems, Mathematics and its Applications, vol. 469, Kluwer Academic Publishers, Dordrecht, 1999. MR 1753470
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- V. Korolyuk and A. Swishchuk, Semi-Markov random evolutions, Mathematics and its Applications, vol. 308, Kluwer Academic Publishers, Dordrecht, 1995. Translated from the 1992 Russian original by V. Zayats and revised by the authors. MR 1472977
- A. V. Skorokhod, D. V. Gusak, A. F. Turbin, and M. N. Kreknin, Vladimir Semenovich Korolyuk, Biobibliografiya Uchenykh Ukrainskoĭ SSR. [Biobibliography of Scientists of the Ukrainian SSR], “Naukova Dumka”, Kiev, 1985 (Russian). MR 874988
- V. S. Koroljuk and A. F. Turbīn, Matematicheskie osnovy fazovogo ukrupneniya slozhnykh sistem, “Naukova Dumka”, Kiev, 1978 (Russian). MR 515139
- I. N. Kovalenko, On the class of limit distributions for thinning streams of homogeneous events, Litovsk. Mat. Sb. 5 (1965), 569–573 (Russian, with Lithuanian and English summaries). MR 0198572
- Igor N. Kovalenko, Rare events in queueing systems—a survey, Queueing Systems Theory Appl. 16 (1994), no. 1-2, 1–49. MR 1272440, DOI https://doi.org/10.1007/BF01158947
- D. S. Sil′vestrov, Predel′nye teoremy dlya slozhnykh sluchaĭ nykh funktsiĭ, Izdat. ObЪed. “Višča Škola” pri Kiev. Gosudarstv. Univ., Kiev, 1974 (Russian). MR 0415731
- D. S. Sīl′vestrov, Polumarkovskie protsessy s diskretnym mnozhestvom sostoyaniĭ, “Sovet. Radio”, Moscow, 1980 (Russian). Biblioteka Inzhenera po Nadezhnosti. [Library for the Engineer in Reliability]; Osnovy rascheta funktsional′nykh i nadezhnostnykh kharakteristik stokhasticheskikh sistem. [Principles of the calculation of functional and reliability characteristics of stochastic systems]. MR 612026
- Dmitrii S. Silvestrov, Limit theorems for randomly stopped stochastic processes, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2004. MR 2030998
- D. Silvestrov, Necessary and sufficient conditions for convergence of first-rare-event times for perturbed semi-Markov processes, Research Report 2016-4, Department of Mathematics, Stockholm University, arXiv:1603.04344, 2016.
- D. S. Silvestrov and M. O. Drozdenko, Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. I, Theory Stoch. Process. 12 (2006), no. 3-4, 151–186. MR 2316558
- D. Silvestrov and S. Silvestrov, Asymptotic expansions for stationary distributions of perturbed semi-Markov processes, Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization (S. Silvestrov and M. Ranc̆ić, eds.), Chapter 10, Springer Proceedings in Mathematics & Statistics, vol. 179, Springer, Heidelberg, 2016, pp. 145–214.
- D. S. Sil′vestrov and Yu. A. Velikiĭ, Necessary and sufficient conditions for the convergence of attainment times, Stability problems for stochastic models (Russian) (Sukhumi, 1987) Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1988, pp. 129–137 (Russian). Translated in J. Soviet Math. 57 (1991), no. 4, 3317–3324. MR 1079133
- A. V. Skorohod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0182056
- A. V. Skorokhod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, 2nd ed., Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics], “Nauka”, Moscow, 1986 (Russian). MR 860563
References
- V. V. Anisimov, Random Processes with Discrete Components, “Vysshaya Shkola”/“Izdatel’stvo Kievskogo Universiteta”, Kiev, 1988. MR 955492
- V. V. Anisimov, Switching Processes in Queueing Models, Applied Stochastic Methods Series, ISTE/Wiley, London/Hoboken, NJ, (2008). MR 2437051
- S. Asmussen, Applied Probability and Queues, Second edition, Applications of Mathematics, vol. 51, Stochastic Modelling and Applied Probability, Springer, New York, 2003. MR 1978607
- S. Asmussen and H. Albrecher, Ruin Probabilities, Second edition, Advanced Series on Statistical Science & Applied Probability, vol. 14, World Scientific, Hackensack, NJ, 2010. MR 2766220
- V. E. Bening and V. Yu. Korolev, Generalized Poisson Models and their Applications in Insurance and Finance, Modern Probability and Statistics, VSP, Utrecht, 2002.
- M. Drozdenko, Weak convergence of first-rare-event times for semi-Markov processes, Theory Stoch. Process. 13(29) (2007), no. 4, 29–63; 15(31) (2009), no. 2, 99–118. MR 2598530
- B. V. Gnedenko and V. Yu. Korolev, Random Summation. Limit Theorems and Applications, CRC Press, Boca Raton, FL, 1996. MR 1387113
- M. Gyllenberg and D. S. Silvestrov, Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems, De Gruyter Expositions in Mathematics, vol. 44, Walter de Gruyter, Berlin, 2008. MR 2456816
- V. V. Kalashnikov, Geometric Sums: Bounds for Rare Events with Applications, Mathematics and its Applications, vol. 413, Kluwer, Dordrecht, 1997. MR 1471479
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, 1996. MR 1451375
- J. Keilson, Markov Chain Models—Rarity and Exponentiality, Applied Mathematical Sciences, vol. 28, Springer, New York, 1979. MR 528293
- D. V. Korolyuk and D. S. Silvestrov, Entry times into asymptotically receding domains for ergodic Markov chains, Teor. Veroyatn. Primen. 28 (1983), 410–420. MR 700223; English transl. in Theory Probab. Appl. 28, 432–442.
- D. V. Korolyuk and D. S. Silvestrov, Entry times into asymptotically receding regions for processes with semi-Markov switchings, Teor. Veroyatn. Primen. 29 (1984), 539–544. MR 761139; English transl. in Theory Probab. Appl. 29, 558–563.
- V. S. Korolyuk, On asymptotical estimate for time of a semi-Markov process being in the set of states, Ukr. Mat. Zh. 21 (1969), 842–845. MR 0251794
- V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Mathematics and its Applications, vol. 469, Kluwer, Dordrecht, 1999. MR 1753470
- V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, Singapore, 2005. MR 2205562
- V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, “Naukova Dumka”, Kiev, 1992; English revised edition, Mathematics and its Applications, vol. 308, Kluwer, Dordrecht, 1995. MR 1472977
- V. S. Korolyuk and A. F. Turbin, Semi-Markov Processes and its Applications, “Naukova Dumka”, Kiev, 1976. MR 874988
- V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of the State Lumping of Large Systems, “Naukova Dumka”, Kiev, 1978. MR 515139; English transl., Mathematics and its Applications, vol. 264, Kluwer, Dordrecht, 1993.
- I. N. Kovalenko, On the class of limit distributions for thinning flows of homogeneous events, Litov. Mat. Sbornik 5 (1965), 569–573. MR 0198572
- I. N. Kovalenko, Rare events in queuing theory—a survey, Queuing Systems Theory Appl. 16 (1994), no. 1–2, 1–49. MR 1272440
- D. S. Silvestrov, Limit Theorems for Composite Random Functions: “Vysshaya Shkola”/“Izdatel’stvo Kievskogo Universiteta”, Kiev, 1974. MR 0415731
- D. S. Silvestrov, Semi-Markov Processes with a Discrete State Space, Library for an Engineer in Reliability, “Sovetskoe Radio”, Moscow, 1980. MR 612026
- D. S. Silvestrov, Limit Theorems for Randomly Stopped Stochastic Processes, Probability and Its Applications, Springer, London, 2004. MR 2030998
- D. Silvestrov, Necessary and sufficient conditions for convergence of first-rare-event times for perturbed semi-Markov processes, Research Report 2016-4, Department of Mathematics, Stockholm University, arXiv:1603.04344, 2016.
- D. S. Silvestrov and M. O. Drozdenko, Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes, Theory Stoch. Process. 12(28) (2006), no. 3–4, 151–186. MR 2316558; 12(28) (2006), no. 3–4, 187–202.
- D. Silvestrov and S. Silvestrov, Asymptotic expansions for stationary distributions of perturbed semi-Markov processes, Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization (S. Silvestrov and M. Ranc̆ić, eds.), Chapter 10, Springer Proceedings in Mathematics & Statistics, vol. 179, Springer, Heidelberg, 2016, pp. 145–214.
- D. S. Silvestrov and Yu. A. Velikii, Necessary and sufficient conditions for convergence of attainment times, Stability Problems for Stochastic Models (V. M. Zolotarev and V. V. Kalashnikov, eds.), Trudy Seminara, VNIISI, Moscow, 1988, pp. 129–137; English transl. in J. Soviet. Math. 57 (1991), 3317–3324. MR 1079133
- A. V. Skorokhod, Random Processes with Independent Increments, Probability Theory and Mathematical Statistics, “Nauka”, Moscow, 1964. MR 0182056; English transl., Nat. Lending Library for Sci. and Tech., Boston, Spa, 1971.
- A. V. Skorokhod, Random Processes with Independent Increments, Second edition, Probability Theory and Mathematical Statistics, “Nauka”, Moscow, 1986. MR 860563; English transl., Mathematics and its Applications, vol. 47, Kluwer, Dordrecht, 1991.
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60J10,
60J22,
60J27,
60K15,
65C40
Retrieve articles in all journals
with MSC (2010):
60J10,
60J22,
60J27,
60K15,
65C40
Additional Information
D. S. Silvestrov
Affiliation:
Department of Mathematics, Stockholm University, SE-106 81 Stockholm, Sweden
Email:
silvestrov@math.su.se
Keywords:
Semi-Markov process,
first-rare-event time,
first-rare-event-time process,
convergence in distribution,
convergence in Skorokhod J-topology,
necessary and sufficient conditions
Received by editor(s):
September 16, 2016
Published electronically:
February 28, 2018
Article copyright:
© Copyright 2018
American Mathematical Society
