Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $p\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}, and let $\mathcal{H}$ be a \reftext{def:independence-sigma-algebras-2026a}{sub-$\sigma$-algebra} of $\mathcal{F}$. Let $\zeta=(\zeta^{1},\dots,\zeta^{p})$ and $\rho=(\rho^{1},\dots,\rho^{p})$ be tuples of \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variables such that:

\textbf{(i)} each $\zeta^{i}$ is \reftext{def:almost-surely-2026a}{almost surely} equal to an $\mathcal{H}$-measurable random variable;

\textbf{(ii)} $\mathbb{E}[\rho^{i}]=0$ for every $i$, with the \reftext{def:expectation-variance-2026a}{expectation};

\textbf{(iii)} the $\sigma$-algebras $\sigma(\rho^{1},\dots,\rho^{p})$ and $\mathcal{H}$ are independent.

Let $M$ be a real $p\times p$ matrix, and let dot products and matrix actions be the \reftext{def:dot-product-orthogonality-rn-2026a}{dot product} and \reftext{def:matrix-vector-product-2026a}{matrix-vector product} applied componentwise to tuples. Then all three expectations below are defined and finite, and with $C_{\rho}:=\bigl(\operatorname{Cov}(\rho^{i},\rho^{j})\bigr)_{1\le i,j\le p}$ (\reftext{def:covariance-square-integrable-2026a}{covariance}) and the \reftext{def:matrix-trace-2026a}{trace}:

where $\zeta+\rho$ is the componentwise sum.