Boolean entropy and information inequalities reveal limits in probability sums
Entropic Inequalities and Limit Theorems in Boolean Probability
Information Theory
Summary
The paper looks at how a special kind of entropy, Boolean entropy, behaves when you add together certain random things that follow Boolean probability rules. The authors find new inequalities related to this entropy and information, and show how the entropy approaches a specific standard pattern. They also provide detailed formulas describing how quickly this convergence happens under certain mathematical conditions. Their work uses advanced mathematical tools to understand these properties better.
What this means in practice
- •For statistical model designers: Use precise limits on Boolean entropy convergence to improve models involving Boolean independent variables and their sums.
- •For signal processing engineers: Incorporate Boolean entropy inequalities to analyze and design systems involving Boolean-like randomness or signals with discrete structure.
A theory result. No direct application yet.
Authors
Kewei Pan
Abstract
We study Boolean entropy and Fisher information in information inequalities and the Boolean central limit theorem. We prove a fractional Stam inequality for the Boolean Fisher information associated with arbitrary fractional partitions of subset sums, and derive an entropy power inequality for sums of Boolean independent random variables and its weighted version. We then establish a Boolean analogue of Barron's entropy convergence result along the Boolean central limit theorem: Boolean entropy converges to the entropy of the Rademacher law if and only if the initial entropy is finite. Furthermore, under a finite fourth moment, in the Boolean central limit theorem, we obtain \[ Γ(μ_n\mid\mathbf b)= \frac{m_4(μ)-1-m_3(μ)^2}{2}\frac{\log n}{n} +o\!\left(\frac{\log n}{n}\right), \] with an explicit example showing sharpness. For symmetric laws with finite fourth moment and finite Boolean Fisher information, \[ Ψ(μ_n\mid\mathbf b) =2\bigl(m_4(μ)-1\bigr)\frac{\log n}{n} +o\left(\frac{\log n}{n}\right). \] Our approach is based on the Nevanlinna representation of Cauchy transforms.