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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Some optimal bivariate Bonferroni-type bounds


Authors: Janos Galambos and Yuan Xu
Journal: Proc. Amer. Math. Soc. 117 (1993), 523-528
MSC: Primary 60E15; Secondary 60C05
DOI: https://doi.org/10.1090/S0002-9939-1993-1146860-7
MathSciNet review: 1146860
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Abstract: Let $ {A_1},{A_2}, \ldots ,{A_n}$ and $ {B_1},{B_2}, \ldots ,{B_m}$ be two sets of events on a probability space. Let $ {X_n}$ and $ {Y_m}$ be the number of those $ {A_j}$ and $ {B_s}$, respectively, that occur. Let $ {S_{k,t}}$ be the $ (k,t){\text{th}}$ binomial moment of the vector $ ({X_n},{Y_m})$. We establish optimal bounds on $ P({X_n} \geqslant 1,{Y_m} \geqslant 1)$ by means of linear combinations of $ {S_{1,1}},\;{S_{2,1}},\;{S_{1,2}}$ and $ {S_{2,2}}$. Optimal lower bounds are also determined when only $ {S_{1,1}},\;{S_{2,1}}$ and $ {S_{1,2}}$ are utilized.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1146860-7
Keywords: Two sets of events, bivariate binomial moments, Bonferroni-type bounds, optimal
Article copyright: © Copyright 1993 American Mathematical Society