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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 $I$?

Federico Poloni
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2 Answers2

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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)

  1. 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

  2. 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

Muralidhar Rangaswamy
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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.

Ross B.
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  • Aren't the variables in the noncentral chi normalized to have variance 1, as from the formula on Wikipedia? I am afraid we need them to have non-1 variance in my question. – Federico Poloni May 08 '13 at 15:12
  • Hm. I think you are right. Let me think about it. – Ross B. May 08 '13 at 19:08
  • @RossB. There are very few mentions of the Noncentral Chi Distribution on the Internet. I have a problem understanding the formulas in the Wikipedia article you've linked. The formula for the mean uses the "generalized Laguerre polynomial", but its n parameter is 1/2 (L^(k/2-1)_1/2), while the linked article only deals with integer values. Can you, please, tell me how to calculate the mean of the noncentral chi distribution? – Ark-kun Sep 30 '14 at 01:17
  • Nevermind. Found the answer here: http://en.wikipedia.org/wiki/Rice_distribution#Moments – Ark-kun Sep 30 '14 at 01:53