Proof of Plane Rotations Preserve Independent Standard Normal Families

The pair $Z_i,Z_j$ is independent (a pair from an independent family is independent, directly from that definition), so by Rotation Invariance of a Pair of Independent Standard Normal Random Variables the random variables $W_i$ and $W_j$ are independent standard normal random variables; and $W_k=Z_k$ is standard normal for every $k\notin\{i,j\}$. It remains to prove that the full family $W_1,\dots,W_m$ is independent.

By the grouping lemma applied to the independent family $(Z_1,\dots,Z_m)$ with the pairwise disjoint blocks $\{i,j\}$ and the singletons $\{k\}$ for $k\notin\{i,j\}$, the $\sigma$-algebras $\sigma(Z_i,Z_j)$ and $\sigma(Z_k)$, $k\notin\{i,j\}$, form an independent family. The random variables $W_i$ and $W_j$ are measurable with respect to $\sigma(Z_i,Z_j)$: the generators $Z_i,Z_j$ are $\sigma(Z_i,Z_j)$-measurable by Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras, and sums and scalar multiples of measurable functions are measurable with respect to any $\sigma$-algebra making the summands measurable, by the preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product.

Fix Borel sets $B_1,\dots,B_m$. The event $A=\{W_i\in B_i\}\cap\{W_j\in B_j\}$ lies in $\sigma(Z_i,Z_j)$, and $\{W_k\in B_k\}=\{Z_k\in B_k\}\in\sigma(Z_k)$ for $k\notin\{i,j\}$. Independence of the family of $\sigma$-algebras gives

$$P\Bigl(A\cap\bigcap_{k\notin\{i,j\}}\{Z_k\in B_k\}\Bigr)=P(A)\prod_{k\notin\{i,j\}}P(Z_k\in B_k),$$

and the independence of $W_i$ and $W_j$ established above gives $P(A)=P(W_i\in B_i)\,P(W_j\in B_j)$. Hence

$$P\Bigl(\bigcap_{k=1}^{m}\{W_k\in B_k\}\Bigr)=\prod_{k=1}^{m}P(W_k\in B_k).$$

The corresponding identity for every subfamily of $W_1,\dots,W_m$ follows by taking $B_k=\mathbb{R}$ for each omitted index, since $P(W_k\in\mathbb{R})=1$. Hence $W_1,\dots,W_m$ are independent in the sense of Independence of Events and of Random Variables. $\blacksquare$