Skip to main content
SpringerLink
Log in

On Bounds for Probabilities of Combinations of Events, the Jordan Formula, and the Bonferroni Inequalities

  • Mathematics
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. N. Frolov, “On inequalities for probabilities wherein at least r from n events occur,” Vestn. St. Petersburg Univ.: Math. 50, 287–296 (2017). https://doi.org/10.3103/S1063454117030074

    Article  MathSciNet  Google Scholar 

  2. A. N. Frolov, “On inequalities for probabilities of joint occurrence of several events,” Vestn. St. Petersburg Univ.: Math. 51, 286–295 (2018). https://doi.org/10.3103/S1063454118030032

    Article  MathSciNet  Google Scholar 

  3. A. N. Frolov, “Bounds for probabilities of unions of events and the Borel-Cantelli lemma,” Stat. Probab. Lett. 82, 2189–2197 (2012).

    Article  MathSciNet  Google Scholar 

  4. 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).

    MathSciNet  MATH  Google Scholar 

  5. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957; Mir, Moscow, 1967), Vol. 1, in Ser.: Wiley Series in Probability and Mathematical Statistics.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. F. M. Hoppe and E. Seneta, “A Bonferroni-type identity and permutation bounds,” Int. Stat. Rew. 58, 253–261 (1990).

    Article  MATH  Google Scholar 

  9. S. Kounias and J. Marin, “Best linear Bonferroni bounds,” SIAM J. Appl. Math. 30, 307–323 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Kounias and K. Sotirakoglou, “Upper and lower bounds for the probability that r events occur,” J. Math. Programming. Oper. Res. 27, 63–78 (1993).

    MathSciNet  MATH  Google Scholar 

  11. E. Margaritescu, “Improved Bonferroni inequalities,” Rev. Roum. Math. Pures Appl. 33, 509–515 (1988).

    MathSciNet  MATH  Google Scholar 

  12. 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).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Prékopa, “Boole-Bonferroni inequalities and linear programming,” Oper. Res. 36, 145–162 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  14. Recsei E and E. Seneta, “Bonferroni-type inequalities,” Adv. Appl. Probab. 19, 508–511 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Sobel and V. R. R. Uppuluri, “On Bonferroni-type inequalities of the same degree for probabilities of unions and intersections,” Ann. Math. Stat. 43, 1549–1558 (1972).

    Article  MATH  Google Scholar 

  16. A. M. Walker, “On the classical Bonferroni inequalities and the corresponding Galambos inequalities,” J. Appl. Probab. 18, 757–763 (1981).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. 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).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. 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).

    Article  MathSciNet  MATH  Google Scholar 

  24. 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). https://doi.org/10.3103/S1063454114020034

    Article  MathSciNet  MATH  Google Scholar 

  25. A. N. Frolov, “On estimation of probabilities of unions of events with applications to the Borel-Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 48, 175–180 (2015). https://doi.org/10.3103/S1063454115030036

    Article  MathSciNet  MATH  Google Scholar 

  26. A. N. Frolov, “On inequalities for conditional probabilities of unions of events and the conditional Borel-Cantelli lemma,” Vestn. St. Petersburg Univ.: Math. 49, 379–388 (2016). https://doi.org/10.3103/S1063454116040063

    Article  MathSciNet  MATH  Google Scholar 

  27. A. N. Frolov, “On inequalities for values of first jumps of distribution functions and Hölder’s inequality,” Stat. Probab. Lett. 126, 150–156 (2017).

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to anonymous reviewers, whose comments favored the improvement of the text of this paper.

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, 199034, Russia

    A. N. Frolov

Authors
  1. A. N. Frolov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Frolov.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frolov, A.N. On Bounds for Probabilities of Combinations of Events, the Jordan Formula, and the Bonferroni Inequalities. Vestnik St.Petersb. Univ.Math. 52, 178–186 (2019). https://doi.org/10.1134/S1063454119020055

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454119020055

Keywords

Search

Navigation