Abstract
New sharp upper and lower bounds for conditional (given a σ-algebra A) probabilities of unions of events and for a generalization of the conditional Borel–Cantelli lemma are obtained. Averaging the left- and right-hand sides of the corresponding inequalities yields bounds better than those obtained by directly estimating the probabilities of events. An example is given. New generalizations of the conditional Borel–Cantelli lemma are also obtained. Averaging yields new versions of this lemma under conditions different from the classical ones.
Similar content being viewed by others
References
A. N. Frolov, “Bounds for probabilities of unions of events and the Borel–Cantelli lemma,” Stat. Probab. Lett. 82, 2189–2197 (2012).
A. N. Frolov, “On inequalities for probabilities of unions of events and the Borel–Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 47, 68–75 (2014).
A. N. Frolov, “On lower and upper bounds for probabilities of unions and the Borel–Cantelli lemma,” Studia Sci. Math. Hungar. 52, 102–128 (2015).
A. N. Frolov, “On estimation for probabilities of unions of events with applications to the Borel–Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 48, 175–180 (2015).
B. L. S. Prakasa Rao, “Conditional independence, conditional mixing and conditional association,” Ann. Inst. Stat. Math. 61, 441–460 (2009).
B. L. S. Prakasa Rao, “Upper and lower bounds for probabilities in the conditional Borel–Cantelli lemma,” Stochastic Anal. Appl. 28, 144–156 (2010).
J. Liu and B. L. S. Prakasa Rao, “On conditional Borel–Cantelli lemmas for sequences of random variables,” J. Math. Anal. Appl. 399, 156–165 (2013).
H.-C. Kim, “On conditional Borel–Cantelli lemma under pairwise extended conditional negative quadrant dependence,” Honam Math. J. 36, 767–775 (2014).
T. K. Chandra, The Borel–Cantelli Lemma (Springer-Verlag, Heidelberg, 2012).
W. Feller, Introduction to the Theory of Probability and Its Applications (Wiley, New York, 1957; Mir, Moscow, 1967).
J. Galambos and I. Simonelli, Bonferroni-Type Inequalities with Applications (Springer-Verlag, New York, 1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.N. Frolov, 2016, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2016, No. 4, pp. 652–663.
About this article
Cite this article
Frolov, A.N. On inequalities for conditional probabilities of unions of events and the conditional Borel–Cantelli lemma. Vestnik St.Petersb. Univ.Math. 49, 379–388 (2016). https://doi.org/10.3103/S1063454116040063
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1063454116040063