Abstract
Necessary and sufficient conditions for P(An infinitely often) = α, α ∈ [0, 1], are obtained, where {An} is a sequence of events such that ΣP(A n ) = ∞.
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References
J. Anděl and V. Dupač, "An extension of the Borel lemma," Comment. Math. Univ. Carolin.,30, No. 2, 403–404 (1989).
P. Erdös and A. Rényi, "On Cantor's series with convergent Σ1/qn," Ann. Univ. Sci. Budapest Eötvös Sect. Math.,2, 93–109 (1959).
R. M. Fischler, "Borel—Cantelli type theorems for mixing sets," Acta Math. Acad. Sci. Hungar.,18, No. 1-2, 67–69 (1967).
S. Kochen and C. Stone, "A note on the Borel—Cantelli lemma," Illinois J. Math.,8, 248–251 (1964).
V. V. Petrov, "An inequality for the moments of a random variable," Teor. Veroyatn. Primenen.,20, No. 2, 402–403 (1975).
V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975).
J. Shuster, "On the Borel—Cantelli problem," Can. Math. Bull.,13, No. 2, 273–275 (1970).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 200–207, 1990.
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Martikainen, A.I., Petrov, V.V. On the Borel-Cantelli lemma. J Math Sci 68, 540–544 (1994). https://doi.org/10.1007/BF01254279
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DOI: https://doi.org/10.1007/BF01254279