## The Annals of Probability

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

### A Multivariate Extension of Hoeffding's Lemma

Henry W. Block and Zhaoben Fang

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62H05: Characterization and structure theory

Secondary: 60E05: Distributions: general theory

**Keywords**

Hoeffding's lemma joint cumulant characterization of independence inequalities for characteristic functions positive dependence association

#### Citation

Block, Henry W.; Fang, Zhaoben. A Multivariate Extension of Hoeffding's Lemma. Ann. Probab. 16 (1988), no. 4, 1803--1820. doi:10.1214/aop/1176991598. https://projecteuclid.org/euclid.aop/1176991598