 StatLect

# Cross-moments of a random vector

This lecture defines the notion of cross-moment of a random vector, which is a generalization of the concept of moment of a random variable (see the lecture entitled Moments of a random variable). ## Cross-moment

Let be a random vector. A cross-moment of is the expected value of the product of integer powers of the entries of : where is the -th entry of and are non-negative integers.

The following is a formal definition of cross-moment.

Definition Let be a random vector. Let and . If exists and is finite, then it is called a cross-moment of of order . If all cross-moments of order exist and are finite, i.e. if (1) exists and is finite for all -tuples of non-negative integers such that , then is said to possess finite cross-moments of order .

The following example shows how to compute a cross-moment of a discrete random vector.

Example Let be a discrete random vector and denote its components by , and . Let the support of be and its joint probability mass function be The following is a cross-moment of of order : which can be computed by using the transformation theorem: ## Central cross-moment

The central cross-moments of a random vector are just the cross-moments of the random vector of deviations .

Definition Let be a random vector. Let and . If exists and is finite, then it is called a central cross-moment of of order . If all central cross-moments of order exist and are finite, that is, if (2) exists and is finite for all -tuples of non-negative integers such that , then is said to possess finite central cross-moments of order .

The following example shows how to compute a central cross-moment of a discrete random vector.

Example Let be a discrete random vector and denote its components by , and . Let the support of be and its joint probability mass function be The expected values of the three components of are The following is a central cross-moment of of order : which can be computed by using the transformation theorem: 