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Conditional independence, conditional mixing and conditional association. (English) Zbl 1314.60054
Summary: Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.

##### MSC:
 60E05 Probability distributions: general theory 60E15 Inequalities; stochastic orderings 60F05 Central limit and other weak theorems 60F15 Strong limit theorems
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