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

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