The fractional Fisher information and the central limit theorem for stable laws

TL;DR

This paper introduces a fractional Fisher information concept to analyze the central limit theorem for stable laws, providing new inequalities and a proof of monotonicity with explicit decay rates.

Contribution

It develops the relative fractional Fisher information and establishes Blachman-Stam type inequalities for stable laws, advancing information-theoretic methods in probability.

Findings

Proposes the concept of relative fractional Fisher information.

Establishes inequalities relating fractional Fisher information of sums.

Provides a proof of monotonicity with explicit decay rates.

Abstract

A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of $n$ independent random variables to the information contained in sums over subsets containing $n-1$ of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate.