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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?

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  • $\begingroup$ what is $\|X\|_2$? $\endgroup$ – Conrado Costa Aug 24 '15 at 19:34
  • $\begingroup$ $\sqrt{\sum_i X_i^2}$ $\endgroup$ – Hedonist Aug 24 '15 at 20:21
  • $\begingroup$ You can use the so called Khinchine-Kahane inequality to show that there are universal constants $c_1,c_2>0$ so that $c_1 \le E\|X\|_2 / \sqrt{E\|X\|_2^2} \le c_2$. $\endgroup$ – Stephen Montgomery-Smith Aug 24 '15 at 21:52
  • $\begingroup$ 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. $\endgroup$ – Hedonist Aug 25 '15 at 12:16
  • $\begingroup$ No, the constants won't depend upon the dimension of the vector. $\endgroup$ – Stephen Montgomery-Smith Aug 25 '15 at 13:44
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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!).

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  • $\begingroup$ 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): math.stackexchange.com/questions/827826/… $\endgroup$ – Hedonist Aug 25 '15 at 17:56
  • $\begingroup$ 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. $\endgroup$ – user357151 Dec 19 '17 at 21:42

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