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Exact conditions for countable inclusion-exclusion identity and extensions. (English) Zbl 1116.05008

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.

MSC:

05A19 Combinatorial identities, bijective combinatorics
05A20 Combinatorial inequalities
60C05 Combinatorial probability

Citations:

Zbl 0158.16405
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