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I wonder if multivariate normal distribution is the only class of distributions whose random vectors have distributions of the same class after linear transformations? How can one justify it?

Is there a name for such property of distributions? Bodie's Investment calls this "stable".

the normal distribution belongs to a special family of distributions characterized as "stable," because of the following property: When assets with normally distributed returns are mixed to construct a portfolio, the portfolio return also is normally distributed.

Thanks!

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    "uniform distributions in simplices" works for nonsingular linear transformations, but not for singular ones. If you take a uniform distribution on an $n$-simplex and project it on a simplex of lower dimension, the distribution is no longer uniform.2011-09-23

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You seem to be asking for more than the "Stable distribution" scenario, you are thinking about transformations of a multivariate variable:

${\bf y} = {\bf A x} + {\bf b}$

with ${\bf A}, {\bf b}$ arbitrary (${\bf A}$ square), so that the "family" (multivariate density) is preserved. I see several problems to give this a clear-cut answer, even a meaning. No only the "family" concept is rather vague, but also the "multivariate" random variable density familiy: for example, we have a definition for a multivariate gaussian, but we don't have (in general) a definition of (say) a multivariate Cauchy. Hence, its difficult to give a useful characterizaton of families of multivariate distributions.

One rather formal way of attacking it would with characteristic functions. Let $\Phi_X({\bf \omega}) = E[\exp(i {\bf \omega^t X })]$ be the (multimensional) c.f. of $X$ and $H_X({\bf \omega}) = \log(\Phi_X{\bf \omega})$. Then, we have

$H_Y({\bf \omega}) = H_X({\bf A^t} {\bf \omega}) + i {\bf \omega}^t {\bf b}$

Thus, we are seeking families of complex functions $H(\omega)$ (with $H(0)=0$) that are closed under the above transformation. One can see immediately (what one already knew) that the gaussian familiy fits, because in that case $H(\omega)$ is a homegeneus cuadratic, and the transformation preserves the property. But I doubt one can say something more.

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    Robert's reply is of course correct. But it must be understood: "discrete distributions" must be regarded as the full family of all discrete distributions (or, in the alternative, the family of all discrete distributions with $n$ values -atoms-). In this case the logarithm in my transformation is not very useful, it's more easy to work with the CF $\Phi_X({\bf \omega})$ itself: if the distribution is discrete, then the CF is a finite sum of complex exponentials (like a Fourier sum). And it's easy to see that the transformed CF retains this property.2011-10-09
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Discrete (multivariate) distributions form one such class. Discrete distributions with at most $n$ atoms form another.

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I dont know if multivariate normal distribution is the only such class.....there are results in statistics which show that if $X+Y$ and $Y$ are normally distributed and $X$,$y$ are independant,then $X$ is normal...this is bernsteins theorem........there are many such interesting characterisations of normal distribution..