Standard Normal Vectors and the Standard Normal Distribution on ℝᵏ

In the setting of Probability Spaces: Notation and Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $k\in\mathbb{N}$, and let the Gaussian constant $c_{\mathrm{G}}>0$ and the exponential function $\exp$ be as in Standard Normal Distribution.

(i) Standard normal density on $\mathbb{R}^{k}$. The standard normal density on $\mathbb{R}^{k}$ is the function $\varphi_{k}:\mathbb{R}^{k}\to\mathbb{R}$,

$$\varphi_{k}(x)=c_{\mathrm{G}}^{-k}\exp\bigl(-\lVert x\rVert^{2}/2\bigr).$$

(ii) Standard normal distribution on $\mathbb{R}^{k}$. The standard normal distribution on $\mathbb{R}^{k}$ is the measure $\gamma_{k}=\varphi_{k}\lambda_{k}$ on $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ with density $\varphi_{k}$ with respect to $\lambda_{k}$ ($\varphi_{k}$ is continuous, since $x\mapsto\lVert x\rVert$ is Lipschitz by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (v), hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous (ii), $\exp$ is continuous by Basic Properties of the Exponential Function (viii), and products, constant multiples and composites of continuous maps are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v) and Continuity of Identity Maps, Compositions and Reciprocals (ii); it is positive by Basic Properties of the Exponential Function (iii) and $c_{\mathrm{G}}>0$; hence it is in $\mathcal{M}^{+}(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv) and Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii), and $\gamma_{k}$ is a measure by Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (v); $\lambda_{k}$ is taken under the axiom of dependent choice).

(iii) Standard normal vectors. A random vector $Z=(Z_{1},\dots,Z_{k})$ in $\mathbb{R}^{k}$ is a standard normal vector if its coordinates $Z_{1},\dots,Z_{k}$ are independent standard normal random variables.