A simplified and unified generalization of some majorization results

TL;DR

This paper unifies and simplifies majorization results for stochastic operators on finite-dimensional $L^1$ spaces, connecting matrix and multivariate majorization with convex inequalities.

Contribution

It provides a unified framework for majorization results, showing approximation of stochastic operators by integral operators and clarifying relationships between different majorization concepts.

Findings

Finite-dimensional stochastic operators can be approximated by integral operators.

Matrix and multivariate majorization are equivalent in $\

,

Abstract

We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in $L^1$). We collect a number of results for vector-valued functions on $L^1$, simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equivalent to convex function inequalities.