Proof of Orthonormal Linear Combinations of Independent Standard Normal Random Variables

For $v\in\mathbb{R}^{m}$ write $v\cdot Z=\sum_{j=1}^{m}v_jZ_j$, a random variable by the closure preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product, so that $W_i=w_i\cdot Z$ with the dot product of $\mathbb{R}^{m}$.

Step 1: Transport along compositions of plane rotations. Assume $m\ge2$ and let $g$ be a plane rotation of $\mathbb{R}^{m}$. If $Y=(Y_1,\dots,Y_m)$ is an independent standard normal family, write $g(Y)$ for the family whose value at $\omega$ is $g\bigl(Y(\omega)\bigr)$; by Plane Rotations Preserve Independent Standard Normal Families, $g(Y)$ is again an independent standard normal family. Moreover, applying Plane Rotations Preserve the Dot Product pointwise with $u=Y(\omega)$,

$$v\cdot Y=g(v)\cdot g(Y)\qquad\text{pointwise on }\Omega,\ \text{for every }v\in\mathbb{R}^{m}.$$

Iterating both facts along a finite composition of plane rotations $h=g_L\circ\dots\circ g_1$, with $h(Y)$ defined by successive application: $h(Y)$ is an independent standard normal family and $v\cdot Y=h(v)\cdot h(Y)$ pointwise, for every $v\in\mathbb{R}^{m}$ (both statements being trivial for the empty composition).

Step 2: The case $m=1$. By part 1 of Alignment of Orthonormal Families by Plane Rotations, $p\le m=1$, so $p=1$, and $w_1=(w_{11})$ with $w_{11}^{2}=1$, so $w_{11}=\pm1$. Then $W_1=Z_1$ or $W_1=-Z_1$, which is standard normal, in the second case by Claim 3 of Reflection Invariance of Lebesgue Measure and Symmetry of the Standard Normal Distribution; a family of one random variable is independent, the product identity being trivial.

Step 3: The case $m\ge2$. By Alignment of Orthonormal Families by Plane Rotations, $p\le m$ and there exist a finite composition $h$ of plane rotations of $\mathbb{R}^{m}$ and $\varepsilon\in\{1,-1\}$ with $h(w_k)=e_k$ for $1\le k\le p-1$ and $h(w_p)=\varepsilon\,e_p$, where $e_1,\dots,e_m$ are the standard basis vectors. Let $Z'=h(Z)$ as in Step 1: an independent standard normal family with $w_k\cdot Z=h(w_k)\cdot Z'$ pointwise for every $k$. Since $e_k\cdot Z'=Z'_k$,

$$W_k=Z'_k\quad(1\le k\le p-1),\qquad W_p=\varepsilon\,Z'_p .$$

If $\varepsilon=1$, then $(W_1,\dots,W_p)=(Z'_1,\dots,Z'_p)$ is a subfamily of an independent standard normal family, hence an independent standard normal family (directly from Independence of Events and of Random Variables).

If $\varepsilon=-1$, then $W_p=-Z'_p$ is standard normal by Claim 3 of Reflection Invariance of Lebesgue Measure and Symmetry of the Standard Normal Distribution, and $W_1,\dots,W_{p-1}$ are standard normal as before. For Borel sets $B_1,\dots,B_p$, the event $\{-Z'_p\in B_p\}$ equals $\{Z'_p\in-B_p\}$, and $-B_p$ is Borel by Claim 1 of Reflection Invariance of Lebesgue Measure and Symmetry of the Standard Normal Distribution; hence, by the independence of $Z'_1,\dots,Z'_p$,

$$P\Bigl(\bigcap_{k=1}^{p-1}\{W_k\in B_k\}\cap\{W_p\in B_p\}\Bigr)=\prod_{k=1}^{p-1}P(Z'_k\in B_k)\cdot P(Z'_p\in-B_p)=\prod_{k=1}^{p}P(W_k\in B_k),$$

using $P(Z'_p\in-B_p)=P(-Z'_p\in B_p)=P(W_p\in B_p)$. The identity for every subfamily follows by taking $B_k=\mathbb{R}$ for omitted indices. Hence $W_1,\dots,W_p$ are independent in the sense of Independence of Events and of Random Variables. $\blacksquare$