In playing with some math finance stuff I ran into the following distribution and I was curious if someone had a name for it or has studied it or worked with it already.

To start, let $\Delta^n$ be the $n$-simplex represented as the points $(x_0,x_1,\ldots,x_n)\in\mathbb{R}^{n+1}$ where $x_k\geq 0$ and $\sum_k x_k=1$. The interior of this set has a natural vector space structure with addition given by $(x_0,\ldots,x_n)+(y_0,\ldots,y_n)=\frac{1}{\sum_k x_ky_k}(x_0y_0,\ldots,x_ny_n)$ and multiplication $\alpha\cdot(x_0,\ldots,x_n)=\frac{1}{\sum_k x_k^\alpha}(x_0^\alpha,\ldots,x_n^\alpha)$. There is a linear map $L$ from $\mathbb{R}^{n+1}$ to the interior given by $L(x_0,x_1,\ldots,x_n)=\frac{1}{\sum_k e^{x_k}}(e^{x_0},\ldots,e^{x_n})$ with kernel the diagonal $x_0=x_1=\ldots=x_n$.

Take a $\mathbb{R}^{n+1}$-valued Gaussian distribution with mean $m$ and covariance matrix $V$. I want the family of distributions on $\Delta^n$ which are the push forward of one of these Gaussians under the map $L$. Well I want the closure of this family in an appropriate sense, which should include some distributions which are concentrated on the boundary of $\Delta^n$. I'm guessing these contain the Dirichlet distributions in some way, at least the symmetric ones.

So do these already have a name?

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    $\begingroup$ Why does the vector space structure matter? $\endgroup$
    – Matt F.
    Oct 28 at 17:56
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    $\begingroup$ It doesn't, really. It's just a satisfying way to view the simplex and these measures as being isomorphic to Gaussians. The idea is you've got some stochastic returns of some assets, elements of the simplex are possible portfolio allocations between those assets. Looking for the optimum portfolio allocation leads to navigating around the simplex, and this is nicely done if you view it as a vector space. $\endgroup$ Oct 28 at 22:38
  • $\begingroup$ @MattF. : Possibly it matters because of the probabilistic interpretation of this "addition" operation as described in my answer below. $\endgroup$ Oct 31 at 14:29

These distributions are heavily relied on in compositional data analysis. See the book by J. Aitchison.

I wrote about the vector-space structure in "Entropies of Likelihood Functions", in Maximum Entropy and Bayesian Methods, Seattle 1991, edited by C.R. Smith, G.J. Erickson, and P.O. Neudorfer, Kluwer, Boston 1992.

Suppose $(x_1,\ldots,x_n) = (x_i : i=1,\ldots,n) = ( \Pr(A_i) : i=1,\ldots,n )$ and $ (y_1,\ldots, y_n) = (y_i: i =1,\ldots,n) \propto (\Pr(D\mid A_i) : i=1,\ldots ,n). $ Then the "sum" of these two in the vector-space structure that you describe is $(\Pr(A_i\mid D) : i=1,\ldots,n).$


The pushforward measure of a Gaussian distribution in $\mathbb R^{n+1}$ under the map $\mathbb R^{n+1}\ni(x_0,x_1,\ldots,x_n)\mapsto(e^{x_0},\ldots,e^{x_n})$ is called a multivariate lognormal distribution; see e.g. this and, in particular, this.

So, you may want to call your distribution a (self-)normalized multivariate lognormal distribution.


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