7.7. Random Vectors#

We will continue to use the notation of capital letters to denote a random vector. We will specify the space over which the random vector is generated to clarify the dimensionality.

A real random vector $$X$$ takes values in the vector space $$\RR^n$$. A complex random vector $$Z$$ takes values in the vector space $$\CC^n$$. We write

$\begin{split} X = \begin{bmatrix} X_1 \\ \vdots \\ X_n \end{bmatrix}. \end{split}$

The expected value or mean of a random vector is $$\EE(X)$$.

$\begin{split} \EE(X) = \begin{bmatrix} \EE(X_1) \\ \vdots \\ \EE(X_n) \end{bmatrix}. \end{split}$

Covariance-matrix of a random vector:

$\Cov (X) = \EE [(X - \EE(X)) (X - \EE(X))^T] = \EE [X X^T] - \EE[X] \EE[X]^T.$

We will use the symbols $$\mu$$ and $$\Sigma$$ for the mean vector and covariance matrix of a random vector $$X$$. Clearly

$\EE [X X^T] = \Sigma + \mu \mu^T.$

Cross-covariance matrix of two random vectors:

$\Cov (X, Y) = \EE [(X - \EE(X)) (Y - \EE(Y))^T] = \EE [X Y^T] - \EE[X] \EE[Y]^T.$

Note that

$\Cov (X, Y) =\Cov (Y, X)^T.$

The characteristic function is defined as

$\Psi_X(j\omega) = \EE \left ( \exp (j \omega^T X) \right ), \quad \omega \in \RR^n.$

The MGF is defined as

$M_X(t) = \EE \left ( \exp (t^T X) \right ), \quad t \in \CC^n.$

Theorem 7.23

The components $$X_1, \dots, X_n$$ of a random vector $$X$$ are independent if and only if

$\Psi_X(j\omega) = \prod_{i=1}^n \Psi_{X_i}(j\omega_i), \quad \forall \omega \in \RR^n.$