Many discrete and continuous univariate probability distributions possess natural, straightforward and unique multivariate extensions/generalizations: the (negative) Binomial, the Gaussian/normal, the Student t, the Beta, the Laplace distribution, etc.

But many univariate probability distributions do no possess such natural and unique extensions: the exponential and Gamma distributions admit quite natural and unique matrix-valued multivariate extensions (the Wishart distribution). But they do not admit a unique and well-defined vector-valued multivariate extension. Same for the Poisson distribution, etc.

For instance, we find at least 23 different multivariate extensions of the exponential distribution:

23 Multivariate exponential distributions and their applications in reliability

And we find also many multivariate extensions of the Poisson distribution:

A Review of Multivariate Distributions for Count Data Derived from the Poisson Distribution

There are many, many papers like that. For instance, we already find more than 5 different bivariate extensions of the exponential distribution, etc.

I’d like to understand why please? To make it concrete:

Why do we talk about THE multivariate Gaussian distribution and why do we talk about 23 multivariate exponential distributions? Conversely, why don't we talk about 23 multivariate Gaussian distributions and why don't we talk about THE multivariate exponential distribution?

What’s the difference between uniquely-multivariatizable distributions and non uniquely-multivariatizable distributions? Can we classify them? What’s the obstruction to unique multivariatization? Is it analytical, algebraic, ...? Feel free to complete both lists.


put on hold as off-topic by Matt F., Denis Serre, RP_, Nik Weaver, LSpice Sep 16 at 21:20

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    $\begingroup$ We can always combine copies of a univariate distribution with a copula. $\endgroup$ – Matt F. Sep 11 at 19:25
  • $\begingroup$ @MattF. Yes, copulas is a well-known way to get multivariate distributions from univariate ones. See e.g. the paper on the Poisson distributions. But my question is not about how we can get, generate multivariate distributions from univariate ones but why sometimes we get a single parametric family of multivariate distributions and why sometimes we can get no less than 23 such parametric families of distributions for the exponential distribution. E.g. we find more than 5 different bivariate extensions for the exponential distribution. $\endgroup$ – Fabrice Pautot Sep 11 at 19:34
  • $\begingroup$ @MattF. I add "unique" multivariatization in the question title to make this point clear. And sorry, "univoque" does not work in English! :( $\endgroup$ – Fabrice Pautot Sep 11 at 19:40
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    $\begingroup$ For me the multiplicity of copulas shows that no distribution has a unique multivariate extension. $\endgroup$ – Matt F. Sep 11 at 19:50
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    $\begingroup$ No, the Gaussian has many multivariate extensions too. Only one is called “the” multivariate Gaussian, but I don’t see any widely applicable criterion of naturality which would justify that article. $\endgroup$ – Matt F. Sep 11 at 20:11