On estimation of probabilities of unions of events with applications to the Borel–Cantelli lemma

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,” Stud. Sci. Math. Hung. 52, 102–128 (2015).

K. L. Chung and P. Erdös, “On the application of the Borel–Cantelli lemma,” Trans. Am. Math. Soc. 72, 179–186 (1952).

S. Gallot, “A bound for the maximum of a number of random variables,” J. Appl. Probab. 3, 556–558 (1966).

D. A. Dawson and D. Sankoff, “An inequality for probabilities,” Proc. Am. Math. Soc. 18, 504–507 (1967).

E. G. Kounias, “Bounds for the probability of a union, with applications,” Ann. Math. Stat. 39, 2154–2158 (1968).

S. M. Kwerel, “Bounds on the probability of the union and intersection of m events,” Adv. Appl. Probab. 7, 431–448 (1975).

S. M. Kwerel, “Most stringent bounds on aggregated probabilities of partially specified probability systems,” J. Am. Stat. Assoc. 70, 472–479 (1975).

J. Galambos, “Bonferroni inequalities,” Ann. Probab. 5, 577–581 (1977).

T. F. Móri and G. J. Székely, “A note on the background of several Bonferroni–Galambos-type inequalities,” J. Appl. Probab. 22, 836–843 (1985).

E. Boros and A. Prékopa, “Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occurs,” Math. Oper. Res. 14, 317–342 (1989).

A. M. Zubkov, “Inequalities for distribution of a number of simultaneously occurring events,” Obozr. Prikl. Prom. Mat. 1, 638–666 (1994).

J. Galambos and I. Simonelli, Bonferroni-type Inequalities with Applications (Springer-Verlag, New York, 1996).

D. de Caen, “A lower bound on the probability of a union,” Discrete Math. 169, 217–220 (1997).

H. Kuai, F. Alajaji, and G. Takahara, “A lower bound on the probability of a finite union of events,” Discrete Math. 215, 147–158 (2000).

V. V. Petrov, “A generalization of the Chung–Erdös inequality for the probability of a union of events,” J. Math. Sci. (New York) 147, 6932–6934 (2007).

A. Prékopa, “Inequalities for discrete higher order convex functions,” J. Math. Inequalities, 485–498 (2009).