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. Alexey M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12 (2006), no. 1-2, 87–93. MR 2316289
  • 2. A. M. Kulik and Yu. N. Kartashov, Invariance principle for additive functionals of Markov chains, arXiv:0704.0508v1.
  • 3. E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
  • 4. A. N. Borodin and I. A. Ibragimov, Limit theorems for functionals of random walks, Trudy Mat. Inst. Steklov. 195 (1994), no. Predel. Teoremy dlya Funktsional. ot Sluchain. Bluzh., 286 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 2 (195) (1995), viii + 259. MR 1368394
  • 5. 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
  • 6. A. N. Kolmogorov and S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, 6th ed., “Nauka”, Moscow, 1989 (Russian). With a supplement, “Banach algebras”, by V. M. Tikhomirov. MR 1025126
    A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 1. Metric and normed spaces, Graylock Press, Rochester, N. Y., 1957. Translated from the first Russian edition by Leo F. Boron. MR 0085462
  • 7. I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
  • 8. A. V. Skorokhod, Lectures on the theory of stochastic processes, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1452108
  • 9. A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405

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