On a strong form of the Borel-Cantelli lemma
DOI:
https://doi.org/10.21638/spbu01.2022.109Abstract
The strong form of the Borel-Cantelli lemma is a variant of the strong law of large numbers for sums of the indicators of events. These sums are centered at the mean and normalized by some function from sums of probabilities of events. The series from probabilities is assumed to be divergent. In this paper, we derive new strong forms of the Borel-Cantelli lemma with smaller normalizing sequences than it was before. Conditions on variations of increments of indicators become stronger. We give examples in which these conditions hold.Keywords:
the Borel-Cantelli lemma, the quantitative Borel-Cantelli lemma, strong forms of the Borel-Cantelli lemma, suns of indicators of events, strong law of large numbers, almost surely convergence
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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 P1/q. Ann. Univ. Sci. Budapest Sect. Math. 2, 93–109 (1959).
3. Spitzer F. Principles of random walk. Princeton, Van Nostrand (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). https://doi.org/10.1016/S0167-7152(02)00113-X
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. Phillipp W. Some metrical theorems in number theory. Pacific J. Math. 20, 109–127 (1967).
9. Петров В.В. О росте сумм индикаторов событий. Записки научных семинаров ПОМИ 298, 150–154 (2003).
10. 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
11. Schmidt W. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110, 493–518 (1964).
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 P1/q. Ann. Univ. Sci. Budapest Sect. Math. 2, 93–109 (1959).
3. Spitzer F. Principles of random walk. Princeton, Van Nostrand (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). https://doi.org/10.1016/S0167-7152(02)00113-X
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. Phillipp W. Some metrical theorems in number theory. Pacific J. Math. 20, 109–127 (1967).
9. 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].
10. 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
11. Schmidt W. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110, 493–518 (1964).
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Published
2022-04-11
How to Cite
Frolov, A. N. (2022). On a strong form of the Borel-Cantelli lemma. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(1), 85–93. https://doi.org/10.21638/spbu01.2022.109
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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.
