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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 80 (2009).
Journal: Theor. Probability and Math. Statist. 80 (2010), 15-23
MSC (2000): Primary 60J55, 60B10, 60F17, 60K10; Secondary 60G46, 60G60
Published electronically: August 18, 2010
MathSciNet review: 2541948
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Abstract | References | Similar Articles | 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.


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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

DOI: http://dx.doi.org/10.1090/S0094-9000-2010-00791-5
PII: S 0094-9000(2010)00791-5
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