Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Borel-Cantelli lemma and its applications

Author: Nuno Luzia
Journal: Trans. Amer. Math. Soc. 366 (2014), 547-560
MSC (2010): Primary 60F05; Secondary 37A30
Published electronically: July 1, 2013
MathSciNet review: 3118406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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.
  • [2] Michael D. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617-631. MR 1231839 (94k:28028),
  • [3] 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 0023010 (9,292f)
  • [4] K. L. Chung and P. Erdös, Probability limit theorems assuming only the first moment. I, Mem. Amer. Math. Soc., 1951 (1951), no. 6, 19. MR 0040612 (12,722g)
  • [5] K. L. Chung and P. Erdös, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc. 72 (1952), 179-186. MR 0045327 (13,567b)
  • [6] 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 (94k:60049),
  • [7] Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153 (98m:60001)
  • [8] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons Inc., New York, 1966. MR 0210154 (35 #1048)
  • [9] Stefano Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett. 14 (2007), no. 5, 797-805. MR 2350125 (2008i:37007)
  • [10] 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.
  • [11] S. Hörmann, On the universal a.s. central limit theorem, Acta Math. Hungar. 116 (2007), no. 4, 377-398. MR 2335804 (2008i:60049),
  • [12] Harry Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447-456. MR 0178494 (31 #2751)
  • [13] 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 (2012g:60105)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 60F05, 37A30

Retrieve articles in all journals with MSC (2010): 60F05, 37A30

Additional Information

Nuno Luzia
Affiliation: Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro 21945-970, Brazil

Keywords: Borel-Cantelli lemma, almost sure local central limit theorem, decay of correlations.
Received by editor(s): January 31, 2012
Received by editor(s) in revised form: July 16, 2012
Published electronically: July 1, 2013
Additional Notes: The author was partially supported by Fundação para a Ciência e a Tecnologia through the project "Randomness in Deterministic Dynamical Systems and Applications" (PTDC/MAT/105448/2008).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society