Generalized Dirichlet distributions on the ball and moments

TL;DR

This paper explores generalized Dirichlet distributions on high-dimensional all geometries, providing new insights into their moments and connections to randomized moment spaces using coordinate transformations.

Contribution

It introduces a novel approach to study Dirichlet distributions on alls, linking geometric and probabilistic perspectives through coordinate changes and moment space analysis.

Findings

Established a new framework for Dirichlet distributions on alls.

Connected randomized all models with randomized moment spaces.

Provided insights into the asymptotic behavior of projections of alls.

Abstract

The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit some known results of this flavour with a new point of view. Roughly speaking, we will endow the ball with some kind of Dirichlet distribution that generalizes the uniform one and will follow the method developed in Skibinsky (1967), Chang et al. (1993) in the context of the randomized moment space. The main idea is to build a suitable coordinate change involving independent random variables. Moreover, we will shed light on a nice connection between the randomized balls and the randomized moment space.