Functional inequalities for Boolean entropy

TL;DR

This paper develops a framework of functional inequalities for Boolean entropy, introducing Boolean Fisher information, Stein discrepancy, and related inequalities, advancing the understanding of Boolean central limit phenomena.

Contribution

It defines Boolean Fisher information and Stein discrepancy, establishes functional inequalities, and derives Berry--Esseen bounds in the Boolean setting, extending classical concepts.

Findings

Boolean Fisher information is monotone in the Boolean CLT

Established a logarithmic Sobolev inequality for Boolean entropy

Derived new Berry--Esseen bounds using Boolean Stein discrepancy

Abstract

Building on the recently introduced notion of Boolean entropy, we define the corresponding Boolean Fisher information via a de Bruijn identity. We study the monotonicity of this Fisher information in the Boolean Central Limit Theorem and establish several functional inequalities involving these quantities, including a logarithmic Sobolev inequality. We also develop Non-microstate counterparts and prove the associated functional inequalities. In addition, we introduce a notion of Stein discrepancy in the Boolean setting, which leads to new Berry--Esseen type bounds in the Boolean central limit theorem.