Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

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



Sufficient conditions for the convergence of local-time type functionals of Markov approximations

Author: Yu. M. Kartashov
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 77 (2007).
Journal: Theor. Probability and Math. Statist. 77 (2008), 39-55
MSC (2000): Primary 60J55, 60J45, 60F17
Published electronically: January 14, 2009
MathSciNet review: 2432771
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A sufficient condition is obtained for the weak convergence of additive functionals defined on a sequence of Markov chains $ X_n$ approaching a Markov process $ X$. The condition is expressed in terms of transient probabilities of the chains $ X_n$. An application of the main result is given for the convergence on the Cantor set of local-time type functionals of random walks approaching an $ \alpha$-stable process with index $ \alpha\leq1$.

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

  • 1. A. M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Proccess. 12(28) (2006), no. 1-2, 87-93. MR 2316289 (2008j:60082)
  • 2. A. M. Kulik and Yu. N. Kartashov, Invariance principle for additive functionals of Markov chains, arXiv:0704.0508v1.
  • 3. E. B. Dynkin, Markov Processes, Fizmatgiz, Moscow, 1963; English transl., vols. I and II, Academic Press and Springer-Verlag, New York and Berlin-Göttingen-Heidelberg, 1965. MR 0193670 (33:1886)
  • 4. A. N. Borodin and I. A. Ibragimov, Limit theorems for functionals of random walks, Trudy Mat. Inst. Steklov., vol. 195, 1994; English transl. in Proc. Steklov Inst. Math. 195 (1995), no. 2. MR 1368394 (97j:60140)
  • 5. W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed., vol. 2, John Wiley & Sons, New York-London-Sydney, 1971. MR 0270403 (42:5292)
  • 6. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, 6th ed., ``Nauka'', Moscow, 1989; English transl. of 1st ed., Graylock, Rochester, NY, 1957. MR 1025126 (90k:46001); MR 0085462 (19:44d)
  • 7. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, ``Nauka'', Moscow, 1965; English transl., Wolters-Noordhoff, Groningen, 1971. MR 0322926 (48:1287)
  • 8. A. V. Skorokhod, Lectures on the Theory of Stochastic Processes, ``Lybid'', Kyiv, 1990; English transl., VSP and TViMS Scientific Publishers, Utrecht and Kiev, 1996. MR 1452108 (99d:60001)
  • 9. A. N. Shiryaev, Probability, ``Nauka'', Moscow, 1980; English transl., Springer-Verlag, New York, 1996. MR 1368405 (97c:60003)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60J55, 60J45, 60F17

Retrieve articles in all journals with MSC (2000): 60J55, 60J45, 60F17

Additional Information

Yu. M. Kartashov
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Keywords: Additive functional, characteristic of an additive functional, Markov approximation
Received by editor(s): May 17, 2007
Published electronically: January 14, 2009
Additional Notes: The work is supported by the Ministry of Science and Education of Ukraine, Project N GP/F13/0095
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society