## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 16, Number 4 (1988), 1803-1820.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991598

Digital Object Identifier
doi:10.1214/aop/1176991598

Mathematical Reviews number (MathSciNet)
MR958217

Zentralblatt MATH identifier
0651.62042

JSTOR