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

Author:
T. I. Kosenkova

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

Journal:
Theor. Probability and Math. Statist. **86** (2013), 123-136

MSC (2010):
Primary 60J25, 60F17, 60B10

DOI:
https://doi.org/10.1090/S0094-9000-2013-00893-X

Published electronically:
August 20, 2013

MathSciNet review:
2986454

Full-text PDF

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.

**1.**Monroe D. Donsker,*An invariance principle for certain probability limit theorems*, Mem. Amer. Math. Soc.,**No. 6**(1951), 12. MR**0040613****2.**Alexey M. Kulik,*Markov approximation of stable processes by random walks*, Theory Stoch. Process.**12**(2006), no. 1-2, 87–93. MR**2316289****3.**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****4.**E. B. Dynkin,*\cyr Markovskie protsessy*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR**0193670****5.**OleksīĭM. 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**, https://doi.org/10.1090/S0094-9000-09-00765-0**6.**A. V. Skorohod,*\cyr Issledovaniya po teorii sluchaĭnykh protsessov (Stokhasticheskie differentsial′nye uravneniya i predel′nye teoremy dlya protsessov Markova)*, Izdat. Kiev. Univ., Kiev, 1961 (Russian). MR**0185619**

A. V. Skorokhod,*Studies in the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0185620****7.**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****8.**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****9.**A. N. Shiryaev,*Probability*, third edition, vol. 1, MCNMO, Moscow, 2004; English transl. of second Russian edition, Springer-Verlag, Berlin-New York, 1996.**10.**A. V. Skorohod,*Limit theorems for stochastic processes*, Teor. Veroyatnost. i Primenen.**1**(1956), 289–319 (Russian, with English summary). MR**0084897****11.**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****12.**I. I. Gikhman and A. V. Skorokhod,*Stochastic Differential Equations*, ``Naukova Dumka'', Kiev, 1968; English transl., Springer-Verlag, New York, 1972.**13.**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**

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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

Email:
tanya.kosenkova@gmail.com

DOI:
https://doi.org/10.1090/S0094-9000-2013-00893-X

Keywords:
L\'evy 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