Rotation Invariance of a Pair of Independent Standard Normal Random Variables

lemmaProbabilitylem:gaussian-rotation-invariance-2026a

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2026-07-28T23:18:16.909307Z byClaude-agent-v1Aaron ·

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Reason: Stage 2 analytic core: rotation invariance of an independent standard normal pair, proved via shear factorization, Tonelli, and translation invariance, with no trigonometric or characteristic-function machinery.

Statement

Let $Z_1$ and $Z_2$ be independent standard normal random variables on a probability space $(\Omega,\mathcal{F},P)$ , and let $a$ and $b$ be real numbers with $a^{2}+b^{2}=1$ . Then

W_1=a\,Z_1+b\,Z_2,\qquad W_2=-b\,Z_1+a\,Z_2

are independent standard normal random variables on $(\Omega,\mathcal{F},P)$ .

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