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October, 1988 A Multivariate Extension of Hoeffding's Lemma
Henry W. Block, Zhaoben Fang
Ann. Probab. 16(4): 1803-1820 (October, 1988). DOI: 10.1214/aop/1176991598


Hoeffding's lemma gives an integral representation of the covariance of two random variables in terms of the difference between their joint and marginal probability functions, i.e., $\operatorname{cov}(X, Y) = \int^\infty_{-\infty} \int^\infty_{-\infty} \{P(X > x, Y > y) - P(X > x)P(Y > y)\} dx dy.$ This identity has been found to be a useful tool in studying the dependence structure of various random vectors. A generalization of this result for more than two random variables is given. This involves an integral representation of the multivariate joint cumulant. Applications of this include characterizations of independence. Relationships with various types of dependence are also given.


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Henry W. Block. Zhaoben Fang. "A Multivariate Extension of Hoeffding's Lemma." Ann. Probab. 16 (4) 1803 - 1820, October, 1988.


Published: October, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0651.62042
MathSciNet: MR958217
Digital Object Identifier: 10.1214/aop/1176991598

Primary: 62H05
Secondary: 60E05

Keywords: association , characterization of independence , Hoeffding's lemma , inequalities for characteristic functions , joint cumulant , Positive dependence

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 4 • October, 1988
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