Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied?
I know that its square follows a generalized chi-square distribution, but this is a different case ("generalized chi"?).
Is there any hope to get expressions for $\|Mv\|$ when $v$ follows a non-Gaussian distribution with mean $0$ and variance $I$?
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$\begingroup$For the case of elliptically contoured distributions (of which the Gaussian is a special case), the distribution of the norm of Mv is available in the literature (See for example 1 and 2)
M. Rangaswamy, D.D. Weiner, and A. Ozturk, "Non Gaussian Random Vector Identification Using Spherically Invariant Random Processes," Aerospace and Electronic Systems, IEEE Transactions on 29 (1), 111-124
Computer generation of correlated non-Gaussian radar clutter M Rangaswamy, D Weiner, A Öztürk
Aerospace and Electronic Systems, IEEE Transactions on 31 (1), 106-116
$\endgroup$ $\begingroup$Yes, the Euclidean norm of a multivariate normal random variable follows a noncentral chi distribution.
Specifically, we have that $y = Mx \sim N(M\mu, M\Sigma M^T)$, so $||y||$ follows a noncentral chi distribution with $k=n$ degrees of freedom and parameter $$\lambda =\sqrt{\sum_i \left(\frac{\mu_{yi}}{\sigma_{yi}}\right)^2}$$
Looking at my nice wall chart of distribution relationships, I can't find an immediate connection to any non-Gaussian variables, but you may have more success.
EDIT: This is incorrect. Working on a fix.
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