Isotropic Gaussian noise has the matching sub-Gaussian proxy

Given: The Axiom of Choice, an integer $n\ge 1$, a parameter $\sigma \ge 0$, and a vector $\epsilon \sim N_n(0,{\sigma }^2{I}_n)$.

Assume the Axiom of Choice, as required by the multivariate-normal construction (The Axiom of Choice).

Under Multivariate normal law, including singular covariance, every linear functional $⟨u,\epsilon ⟩$ has law $N(0,{\sigma }^2\mathrm{\parallel }u{\mathrm{\parallel }}_2^2)$.

A scalar law $N(0,a^2)$ is the law of $aG$ for a standard normal $G$; the standard normal has density $\varphi (x)=(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}$ (Standard normal and normal laws).

Integration against a density gives the corresponding expectation (Integrating against a density agrees with integrating the product).

The standard normal density has total integral one (The standard normal density has total mass one).

Translation is a $C^1$ diffeomorphism and the nonnegative change-of-variables formula applies to its translated Gaussian integrand (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

Axiom of Choice supplies the countable-choice hypothesis required by A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions (AC supplies countable selections and prescribed serial paths).

A centered vector is sub-Gaussian with proxy ${\sigma }^2$ when all its linear functionals satisfy the corresponding exponential-moment bound (Sub-Gaussian variables and vectors, and sub-exponential variables).