Some optimal bivariate Bonferroni-type bounds

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.