A Multivariate Extension of Hoeffding's Lemma

Abstract

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.