Generalized Central Limit Theorem and Renormalization Group

TL;DR

This paper presents a renormalization group approach in probability density space to derive Levy stable laws as fixed points, analyzing their stability and attraction domains, thus providing a physical perspective on the generalized central limit theorem.

Contribution

It introduces a simple renormalization group transformation in probability density space that yields Levy stable laws as fixed points, connecting statistical physics methods with probability theory.

Findings

Levy stable laws are fixed points of the transformation.

The behavior around fixed points depends on the scaling parameter.

The approach offers a physical interpretation of the generalized CLT.

Abstract

We introduce a simple instance of the renormalization group transformation in the Banach space of probability densities. By changing the scaling of the renormalized variables we obtain, as fixed points of the transformation, the L\'evy strictly stable laws. We also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter. The physical interest of a renormalization group approach to the generalized central limit theorem is discussed.