gaussian random vectors are approximately uniform on the hollow sphere

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Take Away

$N (0, I_{d}) \approx Unif (S^{d - 1} (\sqrt{d}))$

Caution

This is an informal approximation!

We get this by noting

When $x \sim N (0, I_{d})$ , we have $Law (\frac{x}{| | x | |}) = Unif (S^{d - 1})$ ( which we get from normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere )
$| | x | | \approx \sqrt{d}$ , which follows from the concentration inequality for magnitude of standard gaussian random vector
Combining the two yields the heuristic approximation.

This idea is formalized in Borel's central limit theorem

gaussian random vectors are approximately uniform on the hollow sphere

References

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