Let's say $X \sim \mathcal{N(0,\Sigma)}$ where $\Sigma \in \mathbb{R}^{n \times n}$. Is there anything we can say about
$$\mathbb{E}_{x\sim X}\left[\sqrt{\sum_i^n x_i^2}\right]$$
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Related: https://math.stackexchange.com/questions/827826/average-norm-of-a-n-dimensional-vector-given-by-a-normal-distribution. – StubbornAtom May 14 '19 at 20:58
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@StubbornAtom Problem is they use a diagonal covariance matrix. I was hoping to have a more general solution. – Armen Aghajanyan May 14 '19 at 21:17
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Google something along the lines of 'expected norm of gaussian vector'. – StubbornAtom May 14 '19 at 21:20
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See, https://mathoverflow.net/questions/155829/euclidian-norm-of-gaussian-vectors – Nap D. Lover May 14 '19 at 21:54
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The question is vague. A crude upper bound can be obtained with Jensen's inequality: $$E(\|X\|_2)\leq \sqrt{E(\|X\|_2^2)}=\sqrt{\sum_{i=1}^n \Sigma_{ii}} = \sqrt{\operatorname{trace}(\Sigma)}$$
Gabriel Romon
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