Citation needed: Only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution

  • #1
DrDu
Science Advisor
6,371
977
TL;DR Summary
I need a citation for the proposition that the only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution.
I need a citation for the following proposition: Assume a random vector ##X=(X_1, ..., X_n)^T## with iid components ##X_i## and mean 0, then the distribution of ##X## is only invariant with respect to orthogonal transformations, if the distribution of the ##X_i## is a normal distribution.
Thank you for your help!
 
Physics news on Phys.org
  • #2
The PDF of [itex]X[/itex] is [tex]
f(x_1)\dots f(x_n)[/tex] where [itex]f[/itex] is the PDF of each [itex]X_i[/itex]. Invariance under orthogonal transformations would require [itex]f[/itex] to be even, since the transformation which multiplies the [itex]i[/itex]th component by -1 and fixes the others is orthogonal. We can then write [itex]f(z) = g(z^2)[/itex] whilst [tex]g(x_1^2) \cdots g(x_n^2) = F(x^TAx)[/tex] for some symmetric matrix [itex]A[/itex] which satisfies [itex]R^TAR = A[/itex] for every orthogonal [itex]R[/itex]. This is equivalent to the requiement that [itex]A[/itex] should commute with every orthogonal [itex]R[/itex]. I believe this in fact results in [itex]A[/itex] being a multiple of the identity. If so, we have [tex]
g(x_1^2) \cdots g(x_n^2) = F(x_1^2 + \dots + x_n^2)[/tex] where the multiplier of the identity has been absorbed into [itex]F[/itex]. Setting all but one of the [itex]x_i[/itex] to be zero then shows that [tex]
g(x_j^2)g(0)^{n-1} = F(x_j^2).[/tex] Setting [itex]g = Ch[/itex] where [itex]h(0) = 1[/itex] we find [itex]
F = C^n h[/itex] where [tex]
h(z_1) \cdots h(z_n) = h(z_1 + \dots + z_n)[/tex] for all [itex](z_1, \dots, z_n) \in [0, \infty)^n[/itex]. I think now we can proceed by induction on [itex]n[/itex], noting that for [itex]n = 2[/itex] and the assumption of continuous [itex]h[/itex] we have [itex]h(z) = h(1)^z = \exp(z\log h(1))[/itex].
 
  • #3
Look up the Maxwell characterization of the multivariate normal distribution.
 

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
896
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
12K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
2K
Back
Top