Entropic Inequalities and Limit Theorems in Boolean Probability

Source 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.