Lévy approximation of an impulse recurrent process with Markov switching
Authors:
V. S. Korolyuk, N. Limnios and I. V. Samoilenko
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 80 (2010), 15-23
MSC (2000):
Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60
DOI:
https://doi.org/10.1090/S0094-9000-2010-00791-5
Published electronically:
August 18, 2010
MathSciNet review:
2541948
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Additional Information
Abstract: The weak convergence of an impulse recurrent process with Markov switching is proved in this paper for the scheme of the Lévy approximation. A modified Liptser semimartingale method is applied to the proof of the relative compactness of the process under consideration. The modification of the Liptser method used in the paper relies upon a solution of a singular perturbation problem instead of an ergodic theorem as used in the original Liptser method.
References
- Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- I. I. Gikhman and A. V. Skorokhod, Teoriya sluchaĭ nykh protsessov. Tom I, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0341539
- Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- R. Sh. Liptser, The Bogolyubov averaging principle for semimartingales, Trudy Mat. Inst. Steklov. 202 (1993), no. Statist. i Upravlen. Sluchaĭn. Protsessami, 175–189 (Russian); English transl., Proc. Steklov Inst. Math. 4(202) (1994), 143–153. MR 1383305
- R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
- Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520
References
- J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, 1996. MR 1406564 (98e:60117)
- S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. MR 838085 (88a:60130)
- I. I. Gikhman and A. V. Skorokhod, Theory of Stochastic Processes, vol. 1, Nauka, Moscow, 1971; English. transl., Springer, Berlin, 1974. MR 0341539 (49:6287)
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987. MR 959133 (89k:60044)
- V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, Singapore, 2005. MR 2205562 (2007a:60004)
- R. Sh. Liptser, The Bogolyubov averaging principle for semimartingales, Trudy Mat. Inst. Steklov., vol. 202, 1993, pp. 175–189; English transl. in Proc. Steklov Inst. Math., vol. 202, 1994, pp. 143–153. MR 1383305 (97c:60117)
- R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, Nauka, Moscow, 1986; English. transl., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. MR 1022664 (90j:60046)
- K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. MR 1739520 (2003b:60064)
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Additional Information
V. S. Korolyuk
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine
Email:
korol@imath.kiev.ua
N. Limnios
Affiliation:
Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, France
Email:
nlimnios@dma.utc.fr
I. V. Samoilenko
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine
Email:
isamoil@imath.kiev.ua
Keywords:
Lévy approximation,
semimartingale,
Markov process,
impulse recurrent process,
piecewise deterministic Markov process,
weak convergence,
singular perturbation
Received by editor(s):
February 9, 2009
Published electronically:
August 18, 2010
Additional Notes:
The authors are grateful to the University of Bielefeld for the hospitality and financial support in the framework of the project DFG 436 UKR 113/80/04-07
Article copyright:
© Copyright 2010
American Mathematical Society
