A Borel-Cantelli lemma and its applications

Luzia, Nuno

doi:10.1090/S0002-9947-2013-06028-X

A Borel-Cantelli lemma and its applications
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by Nuno Luzia PDF

Trans. Amer. Math. Soc. 366 (2014), 547-560 Request permission

Abstract:

We give a version of the Borel-Cantelli lemma. As an application, we prove an almost sure local central limit theorem. As another application, we prove a dynamical Borel-Cantelli lemma for systems with sufficiently fast decay of correlations with respect to Lipschitz observables.

References

J. F. Alves and D. Schnellmann. Ergodic properties of Viana-like maps with singularities in the base dynamics, to appear in Proc. Amer. Math. Soc.
Michael D. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617–631. MR 1231839, DOI 10.1007/BF01244320
Kai-Lai Chung and Paul Erdös, On the lower limit of sums of independent random variables, Ann. of Math. (2) 48 (1947), 1003–1013. MR 23010, DOI 10.2307/1969391
K. L. Chung and P. Erdös, Probability limit theorems assuming only the first moment. I, Mem. Amer. Math. Soc. 6 (1951), 19. MR 40612
K. L. Chung and P. Erdös, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc. 72 (1952), 179–186. MR 45327, DOI 10.1090/S0002-9947-1952-0045327-5
E. Csáki, A. Földes, and P. Révész, On almost sure local and global central limit theorems, Probab. Theory Related Fields 97 (1993), no. 3, 321–337. MR 1245248, DOI 10.1007/BF01195069
Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
Stefano Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett. 14 (2007), no. 5, 797–805. MR 2350125, DOI 10.4310/MRL.2007.v14.n5.a8
N. Haydn, M. Nicol, T. Persson and S. Vaienti. A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, to appear in Ergodic Theory Dynam. Systems.
S. Hörmann, On the universal a.s. central limit theorem, Acta Math. Hungar. 116 (2007), no. 4, 377–398. MR 2335804, DOI 10.1007/s10474-007-6070-1
Harry Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447–456. MR 178494
Michel Weber, A sharp correlation inequality with application to almost sure local limit theorem, Probab. Math. Statist. 31 (2011), no. 1, 79–98. MR 2804977