Entropy inequalities for stable densities and strengthened central limit theorems

TL;DR

This paper introduces an information-theoretic approach to the central limit theorem for stable laws using relative fractional Fisher information, establishing decay rates and inequalities for convergence to stable densities.

Contribution

It develops a new method based on fractional Fisher information to prove explicit convergence rates in the CLT for stable laws, including a fractional logarithmic Sobolev inequality.

Findings

Decay of entropy functionals along the sequence

Explicit convergence rates in various norms

Fractional Fisher information satisfies a logarithmic Sobolev inequality

Abstract

We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a L\'evy centered density of parameter $\lambda >1$ . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced by the same author (cf. arXiv 1504.07057). In particular, it is proven that, with respect to the relative fractional Fisher information, the L\'evy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate.