Randomly Weighted Averages: A Multivariate Case

TL;DR

This paper investigates stochastic linear combinations of random vectors with Dirichlet-distributed coefficients, providing a new, simplified method for calculating their distributions and generalizing previous results in the multivariate case.

Contribution

It introduces a generalized approach for deriving the distribution of Dirichlet-weighted sums of random vectors with a simpler proof than existing methods.

Findings

Derived a new formula for the distribution of Dirichlet-weighted sums

Generalized previous results to multivariate cases

Provided a simpler proof for the distribution calculation

Abstract

Stochastic linear combinations of some random vectors are studied where the distribution of the random vectors and the joint distribution of their coefficients are Dirichlet. A method is provided for calculating the distribution of these combinations which has been studied before by some authors. Our main result is a generalization of some existing results with a simpler proof.