# Expected norm of a random Gaussian vector

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?

• what is $\|X\|_2$? – Conrado Costa Aug 24 '15 at 19:34
• $\sqrt{\sum_i X_i^2}$ – Hedonist Aug 24 '15 at 20:21
• 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