Gaussian Random Vectors: Standing Notation

(i) Base. This setting extends Probability Spaces: Notation, Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force. The axiom of dependent choice is assumed.

(ii) Standard normal densities and laws. $\lambda$ is Lebesgue measure on $\mathbb{R}$ and $c_{\mathrm{G}}$ is the Gaussian constant of The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments. $\varphi$ is the standard normal density and $\gamma$ the standard normal distribution on $\mathbb{R}$, and standard normal random variables are as defined there. For $k\in\mathbb{N}$, $\varphi_{k}$ is the standard normal density and $\gamma_{k}$ the standard normal distribution on $\mathbb{R}^{k}$, and standard normal vectors are as defined there. Background: Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation.

(iii) Gaussian vectors and Gaussian laws. Gaussian random vectors, Gaussian representations and jointly Gaussian random variables are as defined there. For $d\in\mathbb{N}$, $m\in\mathbb{R}^{d}$ and positive semidefinite $\Sigma\in\mathcal{S}(d)$, $N(m,\Sigma)$ is the Gaussian law with mean $m$ and covariance matrix $\Sigma$. Background: Affine Transformations of Gaussian Random Vectors are Gaussian, Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector and The Law of a Gaussian Random Vector Is Determined by Its Mean and Covariance Matrix.