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STATLECT > Additional topics in probability theory

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 X be a Kx1 random vector. A cross-moment of X is the expected value of the product of integer powers of the entries of X:[eq1]where X_i is the i-th entry of X and [eq2] are non-negative integers.

The following is a formal definition of cross-moment:

Definition_ Let X be a Kx1 random vector. Let [eq3] and [eq4]. If[eq5]exists and is finite, then it is called a cross-moment of X of order n. If all cross-moments of order n exist and are finite, i.e. if [eq6] exists and is finite for all K-tuples of non-negative integers [eq7] such that [eq8], then X is said to possess finite cross-moments of order n.

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

Example_ Let X be a $3 imes 1$ discrete random vector and denote its components by X_1, X_2 and $X_{3}$. Let the support of X be: [eq9]and its joint probability mass function be:[eq10]The following is a cross-moment of X of order $4$:[eq11]which can be computed using the transformation theorem:[eq12]

Central cross-moment

The central cross-moments of a random vector X are just the cross-moments of the random vector of deviations [eq13]:

Definition_ Let X be a Kx1 random vector. Let [eq3] and [eq4]. If:[eq16]exists and is finite, then it is called a central cross-moment of $X $ of order n. If all central cross-moments of order n exist and are finite, i.e. if [eq17] exists and is finite for all K-tuples of non-negative integers [eq18] such that [eq8], then X is said to possess finite central cross-moments of order n.

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

Example_ Let X be a $3 imes 1$ discrete random vector and denote its components by X_1, X_2 and $X_{3}$. Let the support of X be: [eq20]and its joint probability mass function be:[eq21]The expected values of the three components of X are:[eq22]The following is a central cross-moment of X of order $3$:[eq23]which can be computed using the transformation theorem:[eq24]

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