normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere

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Theorem

If $x \sim N (0, I_{d})$ , then $Law (\frac{x}{| | x | |}) = Unif (S^{d - 1})$

Proof

The result follows immediately from standard gaussian random vectors are orthogonally invariant and the independence proposition for orthogonally invariant distribution on the unit sphere.

References

#flashcards/math/rmt

What is the important takeaway from the statement "normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere" ?
-?-
The direction of a gaussian random vector is uniformly distributed

References

File	Last Modified
gaussian random vectors are approximately uniform on the hollow sphere	2025-09-11
Random Matrix Lecture 01	2025-09-11
Random Matrix Lecture 02	2025-09-11

normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere

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