Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.25 FOURTH-ORDER GAUSSIAN MOMENT

A fourth-order moment for Gaussian random variables is provided in the following theorem, which is useful later in Chapter 12 on adaptive filtering.

Theorem 5.12 Let {X1, X2, X3, X4} be zero-mean jointly Gaussian random variables. Their joint moment is

(5.348)

Proof. Define the Gaussian random vector whose CF is given by (5.109) after substituting RXX and :

(5.349)

where . Because X has zero mean, this function is real valued (it is not a function of j as are most CFs). Typically such exponentials can be analyzed using a Maclaurin series:

(5.350)

where ··· represent high-order terms. The fourth moment is computed as follows:

(5.351)

which after differentiation and substituting , only the third term in (5.350) remains:

(5.352)

The autocorrelation matrix ...