Expected norm of a random Gaussian vector

Question

Let $X$ be a random vector in $\mathbb{R}^n$ whose entries are joint Gaussian with zero mean and covariance matrix $K.$ Is there a closed form expression for $\mathbb{E}||X||_2,$ as there is for the absolute deviation of a standard Gaussian in a 1-dimensional space?

You can use the so called Khinchine-Kahane inequality to show that there are universal constants $c_1,c_2>0$ so that $c_1 \le E|X|_2 / \sqrt{E|X|_2^2} \le c_2$. — Stephen Montgomery-Smith, Aug 24 '15 at 21:52
I believe the constants $c_1,c_2$ will depend on the dimension of the vector. I would like to have dimension free estimates if possible. Thanks. — Hedonist, Aug 25 '15 at 12:16
No, the constants won't depend upon the dimension of the vector. — Stephen Montgomery-Smith, Aug 25 '15 at 13:44

score 1 · Accepted Answer · answered Aug 25 '15 at 13:41

1

If you can settle with a diagonal covariance matrix, then please check "Multidimensional Gaussian Distributions" by Kenneth S. Miller (1964 edition, chapter 2, section 2, RAYLEIGH DISTRIBUTIONS). Otherwise you need to deal with a lot more complicated equations. This reference could be a good start :

"Properties of Generalized Rayleigh Distributions"
L. E. Blumenson and K. S. Miller
The Annals of Mathematical Statistics
Vol. 34, No. 3 (Sep., 1963), pp. 903-910

You can find a copy of this paper at JSTOR (free sign up!).

answered Aug 25 '15 at 13:41

Ali

682

Thanks Ali. My covariance matrices are not diagonal. I would also like to point out a previous post that solves the problem for identity matrices (your answer subsumes it though): http://math.stackexchange.com/questions/827826/average-norm-of-a-n-dimensional-vector-given-by-a-normal-distribution – Hedonist Aug 25 '15 at 17:56
Unfortunately, where the author of the book talks about "diagonal covariant matrix" (prior to Theorem 1 of Ch. 2 sec 2), he then proceeds to write a scalar multiple of identity. – Dec 19 '17 at 21:42

Expected norm of a random Gaussian vector

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