On strong forms of the Borel-Cantelli lemma and dynamical systems with polynomial decays of correlations
DOI:
https://doi.org/10.21638/spbu01.2022.407Abstract
Strong forms of the Borel - Cantelli lemma are variants of the strong law of large numbers for sums of the indicators of events such that the series from probabilities of these events diverges. These sums are centered at means and normalized by some function from means. In this paper, we derive new strong forms of the Borel - Cantelli lemma under wider restrictions on variations of increments of sums than it was done earlier. Strong forms are commonly used for investigations of properties of dynamical systems. We apply our results to describe properties of some measure preserving expanding maps of [0, 1] with a fixed point at zero. Such results can be proved for similar multidimensional maps as well. Keywords: the Borel - Cantelli lemma, the quantitative Borel - Cantelli lemma, strong forms of the Borel - Cantelli lemma, sums of indicators of events, strong law of large num bers, almost surely convergence, dynamical systems, polynomial decay of correlations.Keywords:
the Borel - Cantelli lemma, the quantitative Borel - Cantelli lemma, strong forms of the Borel - Cantelli lemma, sums of indicators of events, strong law of large numbers, almost surely convergence, dynamical systems, polynomial decay of correlations
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References
Литература
1. Chung K.L., Erd˝os P. On the application of the Borel - Cantelli lemma. Trans. Amer. Math. Soc. 72, 179-186 (1952).
2. Erd˝os P., R´enyi A. On Cantor’s series with convergent 1/q. Ann. Univ. Sci. Budapest Sect. Math. 2, 93-109 (1959).
3. Spitzer F. Principles of random walk. Van Nostrand, Princeton (1964).
4. M´ori T.F., Sz´ekely G.J. On the Erd˝os - R´enyi generalization of the Borel - Cantelli lemma. Studia Sci. Math. Hungar. 18, 173-182 (1983).
5. Petrov V.V. A note on the Borel - Cantelli lemma. Statist. Probab. Lett. 58, 283-286 (2002).
6. Frolov A.N. Bounds for probabilities of unions of events and the Borel - Cantelli lemma. Statist. Probab. Lett. 82, 2189-2197 (2012). https://doi.org/10.1016/j.spl.2012/08/002
7. Frolov A.N. On lower and upper bounds for probabilities of unions and the Borel - Cantelli lemma. Studia Sci. Math. Hungarica 52 (1), 102-128. (2015). https://doi.org/10.1556/SScMath.52.2015 /1/1304
8. Schmidt W. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110, 493-518 (1964).
9. Phillipp W. Some metrical theorems in number theory. Pacific J. Math. 20, 109-127 (1967).
10. Петров В.В. О росте сумм индикаторов событий. Записки научных семинаровПОМИ 298, 150-154 (2003).
11. Kim D. The dynamical Borel - Cantelli lemma for interval maps. Discrete Contin. Dyn. Syst. 17 (4), 891-900 (2007).
12. Gupta C., Nicol M., Ott W. A Borel - Cantelli lemma for non-uniformly expanding dynamical systems. Nonlinearity 23 (8), 1991-2008 (2010).
13. Haydn N., Nicol M., Persson T., Vaienti S. A note on Borel - Cantelli lemmas for nonuniformly hyperbolic dynamical systems. Ergodic Theory Dynam. Systems 33 (2), 475-498 (2013). https://doi.org/10.1017/S014338571100099X
14. Luzia N. Borel - Cantelli lemma and its applications. Trans. Amer. Math. Soc. 366 (1), 547-560 (2014). https://doi.org/10.1090/S0002-9947-2013-06028-X
15. Gou¨ezel S. A Borel - Cantelli lemma for intermittent interval maps. Nonlinearity 20 (6), 1491- 1497 (2007).
16. Frolov A.N. On strong forms of the Borel - Cantelli lemma and intermittent interval maps. J. Math. Analysis Appl. 504 (2), 125425 (2021). https://doi.org/10.1016/j.jmaa.2021.125425
17. Sarig O. Subexponential decay of correlations. Invent. Math. 150, 629-653 (2002).
18. Gou¨ezel S. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29-65 (2004).
19. Hu H., Vaienti S. Lower bounds for the decay of correlations in non-uniformly expanding maps. Ergodic Thoery Dynam. Systems 39, 1936-1970 (2019). https://doi.org/10.48550/arXiv.1307.0359
20. Фролов А.Н. Об усиленнойформе леммы Бореля - Кантелли. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 9 (67), вып. 1, 85-93 (2022). https://doi.org/10.21638/spbu01.2022/109
References
1. Chung K.L., Erd˝os P. On the application of the Borel - Cantelli lemma. Trans. Amer. Math. Soc. 72, 179-186 (1952).
2. Erd˝os P., R´enyi A. On Cantor’s series with convergent 1/q. Ann. Univ. Sci. Budapest Sect. Math. 2, 93-109 (1959).
3. Spitzer F. Principles of random walk. Van Nostrand, Princeton (1964).
4. M´ori T.F., Sz´ekely G.J. On the Erd˝os - R´enyi generalization of the Borel - Cantelli lemma. Studia Sci. Math. Hungar. 18, 173-182 (1983).
5. Petrov V.V. A note on the Borel - Cantelli lemma. Statist. Probab. Lett. 58, 283-286 (2002).
6. Frolov A.N. Bounds for probabilities of unions of events and the Borel - Cantelli lemma. Statist. Probab. Lett. 82, 2189-2197 (2012). https://doi.org/10.1016/j.spl.2012/08/002
7. Frolov A.N. On lower and upper bounds for probabilities of unions and the Borel - Cantelli lemma. Studia Sci. Math. Hungarica 52 (1), 102-128. (2015). https://doi.org/10.1556/SScMath.52.2015 /1/1304
8. Schmidt W. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110, 493-518 (1964).
9. Phillipp W. Some metrical theorems in number theory. Pacific J. Math. 20, 109-127 (1967).
10. Petrov V.V. On the growth of sums of the indicators of events. Zapiski Nauchnykh Seminarov POMI 298, 150-154 (2003) (In Russian) [Eng. transl.: J. Math. Sci. 128, 2578-2580 (2005) https://doi.org/10.1007/s10958-005-0205-0].
11. Kim D. The dynamical Borel - Cantelli lemma for interval maps. Discrete Contin. Dyn. Syst. 17 (4), 891-900 (2007).
12. Gupta C., Nicol M., Ott W. A Borel - Cantelli lemma for non-uniformly expanding dynamical systems. Nonlinearity 23 (8), 1991-2008 (2010).
13. Haydn N., Nicol M., Persson T., Vaienti S. A note on Borel - Cantelli lemmas for nonuniformly hyperbolic dynamical systems. Ergodic Theory Dynam. Systems 33 (2), 475-498 (2013). https://doi.org/10.1017/S014338571100099X
14. Luzia N. Borel - Cantelli lemma and its applications. Trans. Amer. Math. Soc. 366 (1), 547-560 (2014). https://doi.org/10.1090/S0002-9947-2013-06028-X
15. Gou¨ezel S. A Borel - Cantelli lemma for intermittent interval maps. Nonlinearity 20 (6), 1491- 1497 (2007).
16. Frolov A.N. On strong forms of the Borel - Cantelli lemma and intermittent interval maps. J. Math. Analysis Appl. 504 (2), 125425 (2021). https://doi.org/10.1016/j.jmaa.2021.125425
17. Sarig O. Subexponential decay of correlations. Invent. Math. 150, 629-653 (2002).
18. Gou¨ezel S. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29-65 (2004).
19. Hu H., Vaienti S. Lower bounds for the decay of correlations in non-uniformly expanding maps. Ergodic Thoery Dynam. Systems 39, 1936-1970 (2019). https://doi.org/10.48550/arXiv.1307.0359
20. Frolov A.N. Оn a strong form of the Borel - Cantelli lemma. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9 (67), iss. 1, 85-93 (2022). https://doi.org/10.21638 /spbu01.2022/109 (In Russian) [Eng. transl.: Vestnik St Petersburg University, Mathematics 55, iss. 1, 64-70 (2022). https://doi.org/10.1134/S1063454122010058].
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Published
2022-12-26
How to Cite
Frolov А. N. (2022). On strong forms of the Borel-Cantelli lemma and dynamical systems with polynomial decays of correlations. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(4), 644–652. https://doi.org/10.21638/spbu01.2022.407
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
