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 ${\mathbb{R}}^n$. A complex random vector $Z$ takes values in the vector space ${\mathbb{C}}^n$. We write

$$\begin{array}{r}X=\left[\begin{array}{c}{X}_1\\ ⋮\\ X_n\end{array}\right].\end{array}$$

The expected value or mean of a random vector is $\mathbb{E}(X)$.

$$\begin{array}{r}\mathbb{E}(X)=\left[\begin{array}{c}\mathbb{E}(X_1)\\ ⋮\\ \mathbb{E}(X_n)\end{array}\right].\end{array}$$

Covariance-matrix of a random vector:

$$\mathrm{Cov}(X)=\mathbb{E}[(X-\mathbb{E}(X))(X-\mathbb{E}(X){)}^T]=\mathbb{E}[X{X}^T]-\mathbb{E}[X]\mathbb{E}[X{]}^T.$$

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

$$\mathbb{E}[X{X}^T]=\mathrm{\Sigma }+\mu {\mu }^T.$$

Cross-covariance matrix of two random vectors:

$$\mathrm{Cov}(X,Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y){)}^T]=\mathbb{E}[X{Y}^T]-\mathbb{E}[X]\mathbb{E}[Y{]}^T.$$

Note that

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

The characteristic function is defined as

$${\mathrm{\Psi }}_X(j\omega )=\mathbb{E}(\exp (j{\omega }^{T}X)),\omega \in {\mathbb{R}}^n.$$

The MGF is defined as

$$M_X(t)=\mathbb{E}(\exp (t^{T}X)),t\in {\mathbb{C}}^n.$$

Theorem 7.23

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

$${\mathrm{\Psi }}_X(j\omega )=\prod _{i=1}^n{\mathrm{\Psi }}_{X_i}(j{\omega }_i),\mathrm{\forall }\omega \in {\mathbb{R}}^n.$$