Standard Normal Vectors: Density, Moments, Orthogonal Invariance and Concatenation

In the setting of Probability Spaces: Notation and Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, let $k,l,r\in\mathbb{N}$, let $\gamma$ be the standard normal distribution on $\mathbb{R}$ and $\gamma_{k}$ the standard normal distribution on $\mathbb{R}^{k}$, and let $Z$ be a random vector in $\mathbb{R}^{k}$.

(i) The standard normal laws are probability measures. $\gamma$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and $\gamma_{k}$ is a probability measure on $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}))$.

(ii) A standard normal vector exists. On the probability space $(\mathbb{R}^{k},\mathcal{B}(\mathbb{R}^{k}),\gamma_{k})$ ((i)), the identity function of $\mathbb{R}^{k}$ is a standard normal vector.

(iii) Density. $Z$ is a standard normal vector if and only if

$$P_{Z}=\gamma_{k}.$$

(iv) Mean and covariance. If $Z$ is a standard normal vector, then $Z\in\mathcal{L}^{2}(P;\mathbb{R}^{k})$ and

$$\mathbb{E}Z=0,\qquad\operatorname{Cov}(Z)=I_{k}.$$

(v) Orthogonal invariance. If $Z$ is a standard normal vector and $Q\in\mathrm{O}(k)$, then $QZ$ is a standard normal vector.

(vi) Matrices with orthonormal rows. If $Z$ is a standard normal vector and $C\in\mathbb{R}^{r\times k}$ satisfies $CC^{\top}=I_{r}$, then $CZ$ is a standard normal vector in $\mathbb{R}^{r}$.

(vii) Concatenation. If $Z$ is a standard normal vector in $\mathbb{R}^{k}$, $W$ is a standard normal vector in $\mathbb{R}^{l}$, and $Z$ and $W$ are independent, then $(Z,W)$ is a standard normal vector in $\mathbb{R}^{k+l}$.