orthogonally invariant distribution on the unit sphere

[[concept]]

[!themes] Topics

Evaluation Error: SyntaxError: Unexpected token '>'

at DataviewInlineApi.eval (plugin:dataview:19027:21)

at evalInContext (plugin:dataview:19028:7)

at asyncEvalInContext (plugin:dataview:19038:32)

at DataviewJSRenderer.render (plugin:dataview:19064:19)

at DataviewJSRenderer.onload (plugin:dataview:18606:14)

at DataviewJSRenderer.load (app://obsidian.md/app.js:1:1182416)

at DataviewApi.executeJs (plugin:dataview:19607:18)

at DataviewCompiler.eval (plugin:digitalgarden:10763:23)

at Generator.next (<anonymous>)

at fulfilled (plugin:digitalgarden:77:24)

Proposition

There exists a unique probability measure supported on $S^{d - 1}$ that is orthogonally invariant. We denote this $Uniform (S^{d - 1})$

Proposition

Suppose that $x$ is orthogonally invariant. Then

$| | x | |$ is independent of $\frac{x}{| | x | |}$
$Law (\frac{x}{| | x | |}) = Uniform (S^{d - 1})$ .

File	Last Modified
normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere	2025-09-09
Random Matrix Lecture 01	2025-09-11

orthogonally invariant distribution on the unit sphere

References

See Also

Mentions