Do these distributions have a name already?

Question

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?

Answer 1

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 $(\Pr(A_i\mid D) : i=1,\ldots,n).$

Answer 2

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.