Skip to Main Content
Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Strong Markov approximation of Lévy processes and their generalizations in a scheme of series

Author: T. I. Kosenkova
Translated by: N. Semenov
Journal: Theor. Probability and Math. Statist. 86 (2013), 123-136
MSC (2010): Primary 60J25, 60F17, 60B10
Published electronically: August 20, 2013
MathSciNet review: 2986454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of the strong Markov approximation that generalizes the notion of the Markov approximation is introduced. We consider an infinitesimal scheme of series that satisfies the assumptions of a Gnedenko theorem. Under these assumptions, we prove that a sequence of step processes constructed from a corresponding random walk is a strong Markov approximation for a Lévy process. The same result is obtained for a sequence of difference approximations of a solution to a stochastic differential equation driven by a Lévy noise.

References [Enhancements On Off] (What's this?)

  • Monroe D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 6 (1951), 12. MR 40613
  • Alexey M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12 (2006), no. 1-2, 87–93. MR 2316289
  • Yuri N. Kartashov and Alexey M. Kulik, Weak convergence of additive functionals of a sequence of Markov chains, Theory Stoch. Process. 15 (2009), no. 1, 15–32. MR 2603167
  • E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
  • Oleksīĭ M. Kulik, Difference approximation of the local times of multidimensional diffusions, Teor. Ĭmovīr. Mat. Stat. 78 (2008), 86–102 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 78 (2009), 97–114. MR 2446852, DOI
  • A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
  • B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975
  • William Feller, An introduction to probability theory and its applications. Vol. II., 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • A. N. Shiryaev, Probability, third edition, vol. 1, MCNMO, Moscow, 2004; English transl. of second Russian edition, Springer-Verlag, Berlin–New York, 1996.
  • A. V. Skorohod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen. 1 (1956), 289–319 (Russian, with English summary). MR 0084897
  • R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358
  • I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl., Springer-Verlag, New York, 1972.
  • Philip E. Protter, Stochastic integration and differential equations, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR 2020294

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60J25, 60F17, 60B10

Retrieve articles in all journals with MSC (2010): 60J25, 60F17, 60B10

Additional Information

T. I. Kosenkova
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

Keywords: Lévy processes, central limit theorem in a scheme of series, strong Markov approximation
Received by editor(s): June 21, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society