Skip to main content

Variants of the Second BCL

  • Chapter
  • First Online:
The Borel-Cantelli Lemma

Part of the book series: SpringerBriefs in Statistics ((BRIEFSSTATIST,volume 2))

  • 1408 Accesses

Abstract

We shall show here that the second Borel–Cantelli lemma holds for a sequence of events which are pairwise independent. Actually, weaker conditions will suffice.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • S. Amghibech, On the Borel-Cantelli lemma and moments. Comment. Math. Univ. Carolin. 47, 669–679 (2006)

    MathSciNet  MATH  Google Scholar 

  • J. Andel, V. Dupač, An extension of the Borel lemma. Comment. Math. Univ. Carolin. 30, 403–404 (1989)

    Google Scholar 

  • P. Billingsley, Probability and Measure, 3rd edn. (Wiley, New York, 1995), Second Edition 1991. First Edition 1986

    Google Scholar 

  • J.R. Blum, D.L. Hanson, L.H. Koopmans, On the strong law of large numbers for a class of stochastic processes. ZWVG 2, 1–11 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • T.K. Chandra, A First Course in Asymptotic Theory of Statistics (Narosa Publishing House Pvt. Ltd., New Delhi, 1999)

    MATH  Google Scholar 

  • T.K. Chandra, Borel-Cantelli lemma under dependence conditions. SPL 78, 390–395 (2008)

    MATH  Google Scholar 

  • Y.S. Chow, H. Teicher, Probability Theory, 3rd edn. (Springer, New York, 1997)

    Book  MATH  Google Scholar 

  • K.L. Chung, A Course in Probability Theory, 3rd edn. (Academic Press, New York, 2001)

    Google Scholar 

  • H. Cohn, On the Borel-Cantelli lemma. IJM 12, 11–16 (1972)

    MATH  Google Scholar 

  • P. Erdös, A problem about prime numbers and the random walk II. IJM 5, 352–353 (1961)

    MATH  Google Scholar 

  • P. Erdös, A. Rényi, On Cantor’s series with convergent \(\sum 1/q_n\). Ann. Univ. Sci. Budapest Eötvós. Sect. Math. 2, 93–09 (1959)

    MATH  Google Scholar 

  • C. Feng, L. Li, J. Shen, On the Borel-Cantelli lemma and its generalization. C.R. Acad. Sci. Paris Ser. I 347, 1313–1316 (2009)

    Google Scholar 

  • R.M. Fischler, The strong law of large numbers for indicators of mixing sets. Acte Math. Acad. Sci. Hung. 18, 71–81 (1967a)

    Article  MathSciNet  MATH  Google Scholar 

  • R.M. Fischler, Borel-Cantelli type theorems for mixing sets. Acta Math. Acad. Sci. Hung. 18, 67–69 (1967b)

    Article  MathSciNet  MATH  Google Scholar 

  • S.H. Hu, X.J. Wong, X.Q. Li, Y.Y. Zhang, Comments on the paper: A bilateral inequality on the Borel-Cantelli lemma. SPL 79, 889–893 (2009)

    Article  MATH  Google Scholar 

  • K. Itô, H.P. McKean Jr, Potentials and random walk. IJM 4, 119–132 (1960)

    MATH  Google Scholar 

  • D. Khoshnevisan, Probability (American Mathematical Society, Providence, 2007)

    MATH  Google Scholar 

  • S. Kochen, C. Stone, A note on the Borel-Cantelli lemma. IJM 8, 248–251 (1964)

    MathSciNet  MATH  Google Scholar 

  • J. Lamperti, Wiener’s test and Markov chains. J. Math. Anal. Appl. 6, 58–66 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • J. Liu, A note on the bilateral inequality for random variable sequence, P.R. China, Technical Report (2011)

    Google Scholar 

  • A.I. Martikainen, V.V. Petrov, On the Borel-Cantelli lemma. Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. 184, 200–207 (1990). (in Russian)

    Google Scholar 

  • T.F. Móri, G.J. Székeley, On the Erdös-Rényi generalization of the Borel-Cantelli lemma. Studia Sci. Math. Hung. 18, 173–182 (1983)

    MATH  Google Scholar 

  • H.P. McKean Jr, A problem about prime numbers and the random walk I. IJM 5, 351 (1961)

    MathSciNet  Google Scholar 

  • J. Ortega, M. Wschebor, On the sequence of partial maxima of some random sequences. Stoch. Process. Appl. 16, 85–98 (1983)

    Article  MathSciNet  Google Scholar 

  • V.V. Petrov, A note on the Borel-Cantelli lemma. SPL 58, 283–286 (2002)

    Article  MATH  Google Scholar 

  • V.V. Petrov, A generalization of the Borel-Cantelli lemma. SPL 67, 233–239 (2004)

    Article  MATH  Google Scholar 

  • W. Phillipp, Some metrical theorems in numnber theory. Pac. J. Math. 20, 109–127 (1967)

    Google Scholar 

  • A. Rényi, On mixing sequences of sets. Acta Math. Acad. Sci. Hung. 1, 215–228 (1958)

    Article  Google Scholar 

  • A. Rényi, On stable sequences of events. Sankhyā Ser. A 25, 293–302 (1963)

    MATH  Google Scholar 

  • A. Rényi, Probability Theory (North-Holland Publishing Co., Amsterdam, 1970), German Edition 1962. French version 1966. New Hungarian Edition 1965

    Google Scholar 

  • L. Song, Borel-Cantelli lemma for capacities P.R. China, Technical Report (2010)

    Google Scholar 

  • F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, 1964)

    MATH  Google Scholar 

  • J.H. van Lint, R.M. Wilson, A Course in Combinatorics, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  • Y.Q. Xie, A bilateral inequality on the Borel-Cantelli lemma. SPL 78, 2052–2057 (2008)

    Article  MATH  Google Scholar 

  • Y.Q. Xie, A bilateral inequality on nonnegative bounded random sequence. SPL 79, 1577–1580 (2009)

    Article  MATH  Google Scholar 

  • J. Yan, A simple proof of two generalized Borel-Cantelli lemmas, in Memorian Paul-Andre Meyer: Seminaire de Probabilitiés XXXIX. Lecture Notes in Mathematics No. 1874. (Springer-Verlag, 2006)

    Google Scholar 

  • K. Yoshihara, The Borel-Cantelli lemma for strong mixing sequences of events and their applications to LIL. Kodai Math. J. 2, 148–157 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2012 The Author(s)

About this chapter

Cite this chapter

Chandra, T.K. (2012). Variants of the Second BCL. In: The Borel-Cantelli Lemma. SpringerBriefs in Statistics, vol 2. Springer, India. https://doi.org/10.1007/978-81-322-0677-4_3

Download citation

Publish with us

Policies and ethics