H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
2026-03-25 12:53:04.1774443184
How much do tails contribute to a Gaussian's total variance?
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Hölder's inequality gives $\frac{1}{n}\int \mathbf{1}_A \|X\|^2 d\mathbb{P} \leq \frac{1}{n}\left[\int \mathbf{1}_A^2d\mathbb{P}\right]^{1/2}\cdot \left[\int \|X\|^4d\mathbb{P}\right]^{1/2}\leq \sqrt{3 \varepsilon}.$