Borel–Cantelli Lemma and Its Generalizations

Chandra, Tapas Kumar; Kolaneci, Fejzi

doi:10.1007/978-3-642-04898-2_151

Tapas Kumar Chandra² &
Fejzi Kolaneci²

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The celebrated Borel–Cantelli Lemma is important and useful for proving the laws of large numbers in the strong form. Consider a sequence of random events {A _n} on a probability space (Ω, { F}, P), and we are interested in the question of whether infinitely many random events occur or if possibly only a finite number of them occur.

The upper limit of the sequence {A _n} is the random event defined by

$$\{{A}_{n}\,i.o.\} = \mathop {\lim \sup }\limits_{n \to \infty }{A}_{n} = \bigcap \limits_{n=1}^{\infty }\, \bigcup \limits_{k=n}^{\infty }{A}_{ k}\,,$$

which occurs if and only if an infinite number of events A _n occur. This i.o. stands for “infinitely often.”

Below we shall use the fact that if {A _n} is a sequence of random events, then

$$P\left ( \bigcup \limits_{n=1}^{\infty }{A}_{ n}\right ) \leq \ \sum \limits_{n=1}^{\infty }P({A}_{ n}).\quad ({_\ast})$$

The Borel–Cantelli Lemma

Lemma 1

If ∑ ^∞_{n = 1} P(A _n) < ∞, then P(lim sup_{n → ∞} A _n) = 0. If the random events A ₁, A...

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References and Further Reading

Balakrishnan N, Stepanov A (2010) Generalization of Borel-Cantelli lemma. Math Sci 35(1), http://www.appliedprobability.org/content.aspx?Group=tms&Page=tmsabstracts
Barndorff-Nielsen O (1961) On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math Scan 9:383–394
MATH MathSciNet Google Scholar
Chandra TK (2008) The Borel–Cantelli lemma under dependence conditions. Stat Probabil Lett 78:390–395
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Chen LHY (1978) A short note on the conditional Borel-Cantelli lemma. Ann Probab 8:699–700
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Erdös P, Rényi A (1959) On Cantor’s series with convergent ∑ 1 ∕ q _n. Ann Univ Sci Budapest Sec Math 2:93–109
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Kochen SB, Stone CJ (1964) A note on the Borel-Cantelli lemma, Illinois. J Math 8(2):248–251
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Petrov VV (1995) Limit theorems of probability theory. Oxford University Press, Oxford
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Serfling RJ (1975) A general Poisson approximation theorem. Ann Prob 3:726–731
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Stepanov A (2006) Generalization of Borel-Cantelli lemma. eprint: arXiv:math/0605007v1
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Authors and Affiliations

Indian Statistical Institute, Calcutta, India
Tapas Kumar Chandra (Professor) & Fejzi Kolaneci (Professor)

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Tapas Kumar Chandra
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Fejzi Kolaneci
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Editors and Affiliations

Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia
Miodrag Lovric

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Chandra, T.K., Kolaneci, F. (2011). Borel–Cantelli Lemma and Its Generalizations. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_151

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DOI: https://doi.org/10.1007/978-3-642-04898-2_151
Published: 02 December 2014
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04897-5
Online ISBN: 978-3-642-04898-2
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering

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