Proof of Uncorrelated Jointly Gaussian Blocks are Independent

Fix a Gaussian representation of the combined vector: $m$ (zero or a natural number), independent standard normal $Z_1,\dots,Z_m$, real numbers $\mu_i,\nu_k$, and, when $m\ge1$, vectors $x_1,\dots,x_d$ and $y_1,\dots,y_q$ in $\mathbb{R}^{m}$ collecting the coefficients, so that, writing $v\cdot Z=\sum_{j=1}^{m}v_jZ_j$,

Step 1: Orthonormal coordinates for each block. Assume $m\ge1$ (the case $m=0$ is noted in Step 4). Apply Gram-Schmidt orthonormalization to $x_1,\dots,x_d$: either all $x_i$ are zero, in which case set $s=0$, or obtain $s\ge1$ and an orthonormal family $e_1,\dots,e_s$ in $\mathbb{R}^{m}$ with the expansion and span properties of that lemma. Similarly obtain $t=0$, or $t\ge1$ and an orthonormal family $f_1,\dots,f_t$ for $y_1,\dots,y_q$.

Step 2: Cross-orthogonality. By Claim 2 of Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector applied to the fixed representation, $\operatorname{Cov}(X_i,Y_k)=\sum_{j=1}^{m}x_{ij}\,y_{kj}=x_i\cdot y_k$ with the dot product, so the hypothesis gives $x_i\cdot y_k=0$ for all $i,k$. If $s\ge1$ and $t\ge1$: by the span property of Gram-Schmidt Orthonormalization of a Finite Family of Vectors, each $e_u$ is a linear combination of the $x_i$ and each $f_v$ of the $y_k$, so bilinearity of the dot product gives $e_u\cdot f_v=\sum_{i,k}c_{ui}\,c'_{vk}\,(x_i\cdot y_k)=0$. Together with the orthonormality of each family, $(e_1,\dots,e_s,f_1,\dots,f_t)$ is an orthonormal family in $\mathbb{R}^{m}$.

Step 3: Independent Gaussian coordinates. Define $\xi_u=e_u\cdot Z$ ($1\le u\le s$) and $\eta_v=f_v\cdot Z$ ($1\le v\le t$). By Orthonormal Linear Combinations of Independent Standard Normal Random Variables, $(\xi_1,\dots,\xi_s,\eta_1,\dots,\eta_t)$ is an independent standard normal family. If $s\ge1$ and $t\ge1$, the grouping lemma applied with the blocks $\{1,\dots,s\}$ and $\{s+1,\dots,s+t\}$ shows that $\mathcal{G}_1=\sigma(\xi_1,\dots,\xi_s)$ and $\mathcal{G}_2=\sigma(\eta_1,\dots,\eta_t)$ are independent $\sigma$-algebras. If $s=0$ (respectively $t=0$), set $\mathcal{G}_1=\{\emptyset,\Omega\}$ (respectively $\mathcal{G}_2=\{\emptyset,\Omega\}$); the pair $\mathcal{G}_1,\mathcal{G}_2$ is then independent for any choice of the other $\sigma$-algebra, since $P(\emptyset\cap A)=0=P(\emptyset)P(A)$ and $P(\Omega\cap A)=P(A)=P(\Omega)P(A)$ for every event $A$.

Step 4: Almost-sure block representatives. Define

with empty sums equal to $0$; in the case $m=0$, likewise take $\widetilde{X}_i=\mu_i$, $\widetilde{Y}_k=\nu_k$, $s=t=0$, and $\mathcal{G}_1=\mathcal{G}_2=\{\emptyset,\Omega\}$ as in Step 3. If $s\ge1$, the expansion property of Gram-Schmidt Orthonormalization of a Finite Family of Vectors and rearrangement of finite sums give, pointwise on $\Omega$,

so $\widetilde{X}_i=\mu_i+x_i\cdot Z$ pointwise and $P(X_i=\widetilde{X}_i)=1$; if $s=0$ then every $x_i$ is zero (or $m=0$) and again $P(X_i=\mu_i)=P(X_i=\widetilde{X}_i)=1$. Each $\widetilde{X}_i$ is $\mathcal{G}_1$-measurable: constants are measurable with respect to every $\sigma$-algebra, the $\xi_u$ are $\sigma(\xi_1,\dots,\xi_s)$-measurable by Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras, and sums and scalar multiples preserve measurability by the preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product. Symmetrically, $P(Y_k=\widetilde{Y}_k)=1$ and $\widetilde{Y}_k$ is $\mathcal{G}_2$-measurable.

Step 5: Null-set transfer. Let $N=\bigcup_{i=1}^{d}\{X_i\ne\widetilde{X}_i\}\cup\bigcup_{k=1}^{q}\{Y_k\ne\widetilde{Y}_k\}$; then $P(N)=0$, as a finite union of events of probability $0$ (countable additivity and monotonicity of the measure $P$). For subsets $A,\widetilde{A}$ of $\Omega$ write $A\,\triangle\,\widetilde{A}=(A\setminus\widetilde{A})\cup(\widetilde{A}\setminus A)$. The family

is a $\sigma$-algebra: $\Omega\,\triangle\,\Omega=\emptyset\subseteq N$; complements satisfy $(\Omega\setminus A)\,\triangle\,(\Omega\setminus\widetilde{A})=A\,\triangle\,\widetilde{A}$; and countable unions satisfy $\bigl(\bigcup_lA_l\bigr)\,\triangle\,\bigl(\bigcup_l\widetilde{A}_l\bigr)\subseteq\bigcup_l\bigl(A_l\,\triangle\,\widetilde{A}_l\bigr)\subseteq N$. For every Borel set $B$ and every $i$, the generator $X_i^{-1}(B)$ of $\sigma(X_1,\dots,X_d)$ lies in $\mathcal{D}$, with $\widetilde{A}=\widetilde{X}_i^{-1}(B)\in\mathcal{G}_1$ and $A\,\triangle\,\widetilde{A}\subseteq\{X_i\ne\widetilde{X}_i\}\subseteq N$. Since $\sigma(X_1,\dots,X_d)$ is the smallest $\sigma$-algebra containing these generators (Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras), $\sigma(X_1,\dots,X_d)\subseteq\mathcal{D}$. Symmetrically, every $A'\in\sigma(Y_1,\dots,Y_q)$ admits $\widetilde{A}'\in\mathcal{G}_2$ with $A'\,\triangle\,\widetilde{A}'\subseteq N$.

Step 6: Conclusion. Let $A\in\sigma(X_1,\dots,X_d)$ and $A'\in\sigma(Y_1,\dots,Y_q)$, and choose $\widetilde{A}\in\mathcal{G}_1$, $\widetilde{A}'\in\mathcal{G}_2$ as in Step 5. From $A\subseteq\widetilde{A}\cup N$ and $\widetilde{A}\subseteq A\cup N$, monotonicity and additivity of $P$ give $P(A)=P(\widetilde{A}\,)$; likewise $P(A')=P(\widetilde{A}')$; and since $(A\cap A')\,\triangle\,(\widetilde{A}\cap\widetilde{A}')\subseteq(A\,\triangle\,\widetilde{A})\cup(A'\,\triangle\,\widetilde{A}')\subseteq N$, also $P(A\cap A')=P(\widetilde{A}\cap\widetilde{A}')$. By the independence of $\mathcal{G}_1$ and $\mathcal{G}_2$ from Step 3,

As $A$ and $A'$ were arbitrary, the $\sigma$-algebras $\sigma(X_1,\dots,X_d)$ and $\sigma(Y_1,\dots,Y_q)$ are independent in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras. $\blacksquare$