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The multi-variate q-Gaussian distribution in $N$ dimensions takes the following form:

$$G_{q,\sigma}(X) = \frac{1}{\sigma^N K_{q,N}}\left( 1 - \frac{1-q}{(N+4)-(N+2)q} \frac{\|X\|^2}{\sigma^2} \right)^{\frac{1}{1-q}}$$

It appears that the marginal distributions are also q-Gaussian but so far I have not been able to integrate the general form nor to assess if the marginal distributions are q-Gaussians with parameters $(q, \sigma)$ or other parameters.

  • I verified that it is normalized to unity.
  • I wrote down an explicit form for the case $N=2$, with $\|X\|^2=x^2+y^2$ and integrated over x (with Mathematica) but I can't recover something useful.

Any pointer would be greatly appreciated.

Cedric H.
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  • Isn't there a definition with $X^T \Sigma^{-1} X$ instead of $|X|^2/\sigma^2,$ where $\Sigma$ is a positive-definite symmetric $N\times N$ matrix? Usually that's done with multivariate Gaussians. $\qquad$ – Michael Hardy Aug 17 '21 at 19:35
  • @MichaelHardy Yes that would be a more general case. What's assumed here is $\Sigma = \sigma^2 I_{NxN}$. – Cedric H. Aug 17 '21 at 19:43
  • ok. Could you type $N\times N$ instead of $NxN$? $\qquad$ – Michael Hardy Aug 17 '21 at 23:08

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