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Dear mathoverflow community,

I tried to find such a parametric distribution family that "forms a semiring", but I failed.

For example, it is common knowledge that Gaussian distribution family forms a semigroup: let us independently sample 2 variables then their sum also has Gaussian distribution.

So, does such a parametric distribution family exist that is closed with respect to addition and multiplication simultaneously?

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  • $\begingroup$ Take your favorite subring of the real numbers and consider the family of all distributions assuming values in this subring? $\endgroup$
    – Seva
    Commented Feb 13, 2019 at 11:16
  • $\begingroup$ The difficulty is in the word "parametric" which I forgot to add in the post title(( Thanks to you it is fixed. $\endgroup$
    – Denis Kolesnikov
    Commented Feb 13, 2019 at 11:23
  • $\begingroup$ The dirac measures $\delta_x$ parametrized by $x \in \mathbb{R}$ would qualify? Or does "parametric" mean something more specific here? $\endgroup$
    – Steve
    Commented Feb 13, 2019 at 12:54
  • $\begingroup$ Formally it satisfies the conditions, but I want to find distributions with non-zero dispersion. My problem originates from tensor decomposition issue. Let us consider the Tucker decomposition for example - what is the connection between noises in Tucker core and factors and noises in tensor elements? en.wikipedia.org/wiki/Tucker_decomposition $\endgroup$
    – Denis Kolesnikov
    Commented Feb 13, 2019 at 13:38
  • $\begingroup$ If I find the family distribution described above, I can model the noises in tensor elements by proper choice of noise for Tucker core and factors. If such a family doesn't exist, my only hope is that the central limit theorem guarantees that independently distributed noises in tensor decomposition factors cause the Gaussian noises on the tensor elements. $\endgroup$
    – Denis Kolesnikov
    Commented Feb 13, 2019 at 13:39

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The 2013 paper "A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables" by Lennard Bondesson is closely related to your question.

It states the the class of Generalized Gamma Convolutions (Definition 1) satisfies the property you are looking for. However, the class is large: Measures in the class are defined by their Laplace transforms which are "parametrized" by both a measure supported on $(0, \infty)$ and a nonnegative constant. So this is probably not what you were hoping for, but this paper and the related string of literature seem to suggest that there is no "simple" parametrized class of distributions that satisfy the desired property.

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