Bivariate Extension of the Method of Polynomials for Bonferroni-Type Inequalities

Let A1, A2, ..., An and B1, B2, ..., BN be two sequences of events. Let mn(A) and mN(B) be the number of those Aj and Bk, respectively, which occur. Set Sk,t for the joint (k, t)th binomial moment of the vector (mn(A),mN(B)). We prove that linear bounds in terms of the Sk,t on the distribution of the vector (mn(A),mN(B)) are universally true if and only if they are valid in a two dimensional triangular array of independent events Aj and Bi with P(Aj) = p and P(Bi) = s for all j and i. This allows us to establish bounds on P(mn(A) = u, mN(B) = v) from bounds on P(mn − u(A) = 0, mN − v(B) = 0). Several new inequalities are obtained by using our method.