A BorelCantelli lemma and its applications
Author:
Nuno Luzia
Journal:
Trans. Amer. Math. Soc. 366 (2014), 547560
MSC (2010):
Primary 60F05; Secondary 37A30
Published electronically:
July 1, 2013
MathSciNet review:
3118406
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Abstract 
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Additional Information
Abstract: We give a version of the BorelCantelli lemma. As an application, we prove an almost sure local central limit theorem. As another application, we prove a dynamical BorelCantelli lemma for systems with sufficiently fast decay of correlations with respect to Lipschitz observables.
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 [1]
 J. F. Alves and D. Schnellmann. Ergodic properties of Vianalike 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, 617631. MR 1231839 (94k:28028), http://dx.doi.org/10.1007/BF01244320
 [3]
 KaiLai Chung and Paul Erdös, On the lower limit of sums of independent random variables, Ann. of Math. (2) 48 (1947), 10031013. 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 BorelCantelli lemma, Trans. Amer. Math. Soc. 72 (1952), 179186. 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, 321337. MR 1245248 (94k:60049), http://dx.doi.org/10.1007/BF01195069
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 Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153 (98m:60001)
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 William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons Inc., New York, 1966. MR 0210154 (35 #1048)
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 Stefano Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett. 14 (2007), no. 5, 797805. MR 2350125 (2008i:37007)
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 N. Haydn, M. Nicol, T. Persson and S. Vaienti. A note on BorelCantelli lemmas for nonuniformly 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, 377398. MR 2335804 (2008i:60049), http://dx.doi.org/10.1007/s1047400760701
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 Harry Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447456. 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, 7998. MR 2804977 (2012g:60105)
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Additional Information
Nuno Luzia
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro 21945970, Brazil
Email:
nuno@im.ufrj.br
DOI:
http://dx.doi.org/10.1090/S00029947201306028X
Keywords:
BorelCantelli 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.
