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

摘要

Let A_1,A_2...,A_n and B_1, B_2,...,B_n be two sequences of events. Let m_n(A) and m_N(B) be the number of thoseA_j and B_k , respectively, which occur. Set S_(k,t), for the joint (k,t)th binomial moment of the vector (m_n(A), m_N(B)), We prove that lineur bounds in terms of the _(k,t), on the distribution of the vector (m_n(A),m_N(B)) are universally true if and only if they are valid in u two dimensional triangular array of independent events A_j and B_i with P(A_j)=p and P(B_i)=s for all j and i. This allows us to establish bounds on P(m_n(A)=u, m_n(B)=v) from bounds on P(m_(n-u)(A)=0, m_(N-v)(B)=0). Several new inequalities are obtained by using our method.