Entropic Symmetrization Resistance

2026-07-31

AI Summary

This study investigates whether asymmetric random variables on locally compact groups exhibit entropy symmetrization resistance—that is, any independent random variable that renders their sum symmetric must necessarily possess strictly greater entropy. Extending the notion of symmetrization resistance from the variance-based framework to the entropy-based setting for the first time, the work establishes a theoretical foundation for this property on finite cyclic groups by integrating tools from information theory, Fourier analysis on groups, and boundary properties of probability simplices. The main contributions include proving that asymmetric Bernoulli distributions in real and higher-dimensional spaces are entropy symmetrization resistant, and providing a complete characterization of all such distributions on ℤ₃ and ℤ₄.

Abstract

An asymmetric random variable $X$ in the reals is said to be variance symmetrization resistant if every independent random variable $Y$ in the reals that produces a symmetric sum $X+Y$ has a greater variance than that of $X$. Asymmetric Bernoulli random variables were shown to be variance symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We introduce the notion of entropic symmetrization resistance on locally compact groups-- this means that the entropy of any independent symmetrizer $Y$ must exceed that of $X$. We show that asymmetric Bernoulli random variables exhibit entropic symmetrization resistance, and show a multidimensional generalization. We also explore basic aspects of the entropic symmetrization resistance problem in compact groups. In particular, we show that any distribution on a finite group that is entropic symmetrization resistant must lie on the boundary of the probability simplex, and describe precisely the class of all entropic symmetrization resistant distributions on $\mathbb{Z}_3$ and $\mathbb{Z}_4$.

Problem

• entropic symmetrization resistance
• asymmetric distributions
• entropy
• locally compact groups
• symmetrizer

Innovation

• entropic symmetrization resistance
• locally compact groups
• asymmetric Bernoulli
• entropy
• probability simplex