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), 123136
MSC (2010):
Primary 60J25, 60F17, 60B10
Published electronically:
August 20, 2013
MathSciNet review:
2986454
Fulltext 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., 1951 (1951),
no. 6, 12. MR 0040613
(12,723a)
 2.
Alexey
M. Kulik, Markov approximation of stable processes by random
walks, Theory Stoch. Process. 12 (2006),
no. 12, 87–93. MR 2316289
(2008j:60082)
 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
(2011a:60130)
 4.
E.
B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.Mat.
Lit., Moscow, 1963 (Russian). MR 0193670
(33 #1886)
 5.
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
(2010b:60212), 10.1090/S0094900009007650
 6.
A.
V. Skorokhod, Studies in the theory of random processes,
Translated from the Russian by Scripta Technica, Inc, AddisonWesley
Publishing Co., Inc., Reading, Mass., 1965. MR 0185620
(32 #3082b)
 7.
B.
V. Gnedenko and A.
N. Kolmogorov, Limit distributions for sums of independent random
variables, AddisonWesley Publishing Company, Inc., Cambridge, Mass.,
1954. Translated and annotated by K. L. Chung. With an Appendix by J. L.
Doob. MR
0062975 (16,52d)
 8.
William
Feller, An introduction to probability theory and its applications.
Vol. II., Second edition, John Wiley & Sons, Inc., New
YorkLondonSydney, 1971. MR 0270403
(42 #5292)
 9.
A. N. Shiryaev, Probability, third edition, vol. 1, MCNMO, Moscow, 2004; English transl. of second Russian edition, SpringerVerlag, BerlinNew 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
(18,943c)
 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
(2003h:60001)
 12.
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, ``Naukova Dumka'', Kiev, 1968; English transl., SpringerVerlag, New York, 1972.
 13.
Philip
E. Protter, Stochastic integration and differential equations,
2nd ed., Applications of Mathematics (New York), vol. 21,
SpringerVerlag, Berlin, 2004. Stochastic Modelling and Applied
Probability. MR
2020294 (2005k:60008)
 1.
 M. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 6 (1951), 110. MR 0040613 (12:723a)
 2.
 A. M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12(28) (2006), no. 12, 8793. MR 2316289 (2008j:60082)
 3.
 Yu. N. Kartashov and A. M. Kulik, Weak convergence of additive functionals of a sequence of Markov chains, Theory Stoch. Process. 15(31) (2009), no. 1, 1532. MR 2603167 (2011a:60130)
 4.
 E. B. Dynkin, Markov processes, ``Fizmatgiz'', Moscow, 1963; English transl., SpringerVerlag, BerlinGöttingenHeidelberg, vol. I and II, 1965. MR 0193670 (33:1886)
 5.
 A. M. Kulik, Difference approximation of the local times of multidimensional diffusions, Teor. Imovir. Matem. Statyst. 78 (2008), 86102; English transl. in Theor. Probability and Math. Statist. 78 (2009), 8395. MR 2446852 (2010b:60212)
 6.
 A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University, Kiev, 1961; English transl., AddisonWesley, Reading, 1965. MR 0185620 (32:3082b)
 7.
 B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, ``Gostehizdat'', Moscow, 1949; English transl., AddisonWesley (Cambridge, MA), 1954. MR 0062975 (16:52d)
 8.
 W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, Inc., New YorkLondonSydney, 1971. MR 0270403 (42:5292)
 9.
 A. N. Shiryaev, Probability, third edition, vol. 1, MCNMO, Moscow, 2004; English transl. of second Russian edition, SpringerVerlag, BerlinNew York, 1996.
 10.
 A. V. Skorokhod, Limit theorems for stochastic processes, Teor. Veroyatnost. Primenen. 1 (1956), 289319; English transl. in Theory Probab. Appl. 1, 261290. MR 0084897 (18:943c)
 11.
 R. M. Dudley, Real Analysis and Probability, Cambridge University Press, New York, 2004. MR 1932358 (2003h:60001)
 12.
 I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, ``Naukova Dumka'', Kiev, 1968; English transl., SpringerVerlag, New York, 1972.
 13.
 P. E. Protter, Stochastic Integration and Differential Equations, Second edition, Springer, New York, 2004. MR 2020294 (2005k:60008)
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
Email:
tanya.kosenkova@gmail.com
DOI:
http://dx.doi.org/10.1090/S00949000201300893X
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
