Friedland, Shmuel; Krop, Elliot Exact conditions for countable inclusion-exclusion identity and extensions. (English) Zbl 1116.05008 Int. J. Pure Appl. Math. 29, No. 2, 177-182 (2006). Let \((\Omega ,\Sigma ,P)\) be a probability space, \(A_1,A_2,\dots ,A_n\in \Sigma \) and \(S_k=\sum _{1\leq i_{1}<\cdots <i_{k}}P(A_{i_{1}}\cap \cdots \cap A_{i_{k}})\) for \(k\in N\). In this paper it is proved that \[ P(\bigcup _{1\leq i_{1}<\cdots <i_{k}}(A_{i_{1}}\cap \cdots \cap A_{i_{k}})=\sum _{j\in Z_{+}}(-1)^{j} {{j+k-1}\choose{k-1}}S_{j+k} \]for some \(k\in N\), if and only if \(S_l\), \(l\in N\), is a sequence of nonnegative numbers such that \(\lim _{l\rightarrow \infty }l^{k-1}S_{l}=0\). In terms of a random variable, whose range are nonnegative integers, this condition is equivalent to the convergence to zero of binomial moments. Some equalities due to L. Takács [J. Am. Stat. Assoc. 62, 102–113 (1967; Zbl 0158.16405)] are extended this way. Reviewer: Ioan Tomescu (Bucureşti) MSC: 05A19 Combinatorial identities, bijective combinatorics 05A20 Combinatorial inequalities 60C05 Combinatorial probability Keywords:Bonferroni inequalities; binomial moments; random variable Citations:Zbl 0158.16405 PDFBibTeX XMLCite \textit{S. Friedland} and \textit{E. Krop}, Int. J. Pure Appl. Math. 29, No. 2, 177--182 (2006; Zbl 1116.05008) Full Text: arXiv