Independent Random Variables and Independent Standard Normal Coordinates on a Product Space

In the setting of Gaussian Random Vectors: Standing Notation and Convergence of Probability Measures and Random Elements: Standing Notation, let $(\Omega,\mathcal{F},P)$ be a probability space, let $n\in\mathbb{N}$, and let $Y_{1},\dots,Y_{n}$ be random variables on it such that the family $(Y_{k})_{k\in[n]}$ is independent. Let

$$(\tilde\Omega,\tilde{\mathcal{F}},\tilde P)=\bigl(\Omega\times\mathbb{R}^{n},\ \mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{n}),\ P\otimes\gamma_{n}\bigr),$$

for $k\in[n]$ let $\tilde Y_{k},\tilde Z_{k}:\tilde\Omega\to\mathbb{R}$ be the functions

$$\tilde Y_{k}(\omega,z)=Y_{k}(\omega),\qquad\tilde Z_{k}(\omega,z)=z_{k},$$

and let $(W_{i})_{i\in[2n]}$ be the family with $W_{k}=\tilde Y_{k}$ and $W_{n+k}=\tilde Z_{k}$ for $k\in[n]$.

(i) The product space. $(\tilde\Omega,\tilde{\mathcal{F}},\tilde P)$ is a probability space, and $\tilde Y_{k}$ and $\tilde Z_{k}$ are random variables on it for every $k\in[n]$.

(ii) Independence. The family $(W_{i})_{i\in[2n]}$ is independent.

(iii) Laws. For every $k\in[n]$, $\tilde Y_{k}$ and $Y_{k}$ are identically distributed, and $\tilde Z_{k}$ is a standard normal random variable.