Limit theorems for difference additive functionals
Author:
Yu. M. Kartashov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 83 (2011), 83-94
MSC (2010):
Primary 60J55, 60J45, 60F17
DOI:
https://doi.org/10.1090/S0094-9000-2012-00843-0
Published electronically:
February 2, 2012
MathSciNet review:
2768850
Full-text PDF Free Access
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Additional Information
Abstract: We consider additive functionals defined on Markov chains that approximate a Markov process. Sufficient conditions are obtained for the convergence of the functionals. These conditions are expressed in terms of convergence of some conditional expectations (called the characteristics of the functionals) under general assumptions on the convergence of processes. Sufficient conditions for the uniform convergence of additive functionals are also given.
References
- 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
- Yu. M. Kartashov, Sufficient conditions for the convergence of functionals of local-time type of Markov approximations, Teor. Ĭmovīr. Mat. Stat. 77 (2007), 36–51 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 77 (2008), 39–55. MR 2432771, DOI https://doi.org/10.1090/S0094-9000-09-00746-7
- Yu. N. Kartashov and A. M. Kulik, Convergence of difference additive functionals under local conditions on their characteristics, Ukraïn. Mat. Zh. 61 (2009), no. 9, 1208–1224 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 61 (2009), no. 9, 1428–1447. MR 2752551, DOI https://doi.org/10.1007/s11253-010-0287-9
- Alexey M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12 (2006), no. 1-2, 87–93. MR 2316289
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Claude Dellacherie and Paul-André Meyer, Probabilities and potential. B, North-Holland Mathematics Studies, vol. 72, North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales; Translated from the French by J. P. Wilson. MR 745449
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- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
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References
- 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, 15–32. MR 2603167 (2011a:60130)
- Yu. M. Kartashov, Sufficient conditions for the convergence of functionals of local-time type of Markov approximations, Teor. Imovir. Mat. Stat. 77 (2007), 36–51; English transl. in Theory Probab. Math. Statist. 77 (2008), 39–55. MR 2432771 (2010a:60272)
- Yu. N. Kartashov and A. M. Kulik, Convergence of difference additive functionals under local conditions on their characteristics, Ukrain. Mat. Zh. 61 (2009), no. 9, 1208–1224; English transl. in Ukrainian Math. J. 61 (2009), no. 9, 1428–1447. MR 2752551
- A. M. Kulik, Markov approximation of stable processes by random walks, Theory Stoch. Process. 12(28) (2006), no. 1–2, 87–93. MR 2316289 (2008j:60082)
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0233396 (38:1718)
- C. Dellacherie and P.-A. Meyer, Probabilities and Potentials, North-Holland Publishing Company, Amsterdam–New York, 1982. MR 745449 (85e:60001)
- R. Bass and D. Khoshnevisan, Local times on curves and uniform invariance principles, Probab. Theory Related Fields 92 (1992), no. 4, 465–492. MR 1169015 (93e:60161)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, Rhode Island, 2000. MR 1743716 (2001g:60089)
- E. B. Dynkin, Markov Processes, Moscow, Fizmatgiz, 1963; English transl., Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1965, Vols. I and II. MR 0193670 (33:1886)
- A. N. Shiryaev, Probability, Moscow, Nauka, 1980; English transl., Springer-Verlag, New York, 1996. MR 1368405 (97c:60003)
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Additional Information
Yu. M. Kartashov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
kartashov-y@yandex.ru
Keywords:
Additive functional,
characteristics of an additive functional,
invariance principle
Received by editor(s):
March 3, 2010
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
