(1d Gaussian r.v.)
Definition: Elementary (or standard) Gaussian random vector is a random vector composed of real, mean-zero, unit variance, independent Gaussian r.v.'s.
According to the definition, the PDF and CF of standard Gaussian random vector \( \mathbf{X} \) are
\[ f_{\mathbf{X}} (\mathbf{x}) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{x_i^2}{2}} = \frac{1}{\sqrt{2\pi}^n} e^{-\frac{1}{2} \mathbf{x}^T \mathbf{x}} \]
\[ \Phi_{\mathbf{X}} (\mathbf{w}) = \prod_{i=1}^n e^{-\frac{w_i^2}{2}} = e^{-\frac{1}{2} \mathbf{w}^T \mathbf{w}} \]
Definition: Any random vector that can be generated by a linear/affine transformation of a standard Gaussian random vector is called a Gaussian random vector.
Theorem: The CF of a Gaussian random vector \( \mathbf{Y} \) with mean \( \mathbf{m}_Y \) and covariance matrix \( K_Y \) is
\[ \Phi_{\mathbf{Y}} (\mathbf{w}) = e^{i \mathbf{w}^T \mathbf{m}_Y -\frac{1}{2} \mathbf{w}^T K_Y \mathbf{w}} \]
Theorem: Linear/Affine transformations of Gaussian random vectors are Gaussian random vectors.
Theorem: The PDF of a Gaussian random vector \( \mathbf{Y} \) with nonsingular \( K_Y \) is
\[ f_{\mathbf{Y}} (\mathbf{y}) = \frac{1}{\sqrt{ (2\pi)^n \det(K_Y)}} e^{-\frac{1}{2} ( \mathbf{y} - \mathbf{m}_Y )^T K_Y^{-1} ( \mathbf{y} - \mathbf{m}_Y )} \]
Theorem: The PDF of a Gaussian random vector \( \mathbf{Y} \) with singular \( K_Y \) is
\[ f_{\mathbf{Y}} (\mathbf{y}) = f_{\mathbf{Y}_a} (\mathbf{y}_a) \delta( \mathbf{y}_b - [\mathbf{m}_b + BA^{-1}(\mathbf{y}_a - \mathbf{m}_a) ] ) \]
Here we denote \( \text{rank}(K_Y) = m \) and suppose the first m elements of Y, \( \mathbf{Y}_a \), have nonsingular covariance matrix. Decompose \( K_Y = E \Lambda E^T = H H^T \), with \( H = ( \sqrt{\lambda_1} \mathbf{e}_1, \cdots, \sqrt{\lambda_m} \mathbf{e}_m) \). Write \( H = \begin{pmatrix} A \\ B \end{pmatrix} \), where A is m-by-m and B is (n-m)-by-m.
Theorem: For any Gaussian random vectors \( \mathbf{X}_1, \mathbf{X}_2 \),
\[ \mathbf{X}_1 ∐ \mathbf{X}_2 \Leftrightarrow \text{Cov}(\mathbf{X}_1, \mathbf{X}_2) = 0 \]
Click here for all the proofs.