# An elementary renormalization-group approach to the Generalized Central Limit Theorem and Extreme Value Distributions

## TL;DR

This paper presents a transparent, renormalization-group-inspired approach to deriving the Generalized Central Limit Theorem and extreme value distributions, highlighting their universality and underlying similarities.

## Contribution

It introduces a non-rigorous but insightful renormalization-group method to derive key limit theorems in probability theory, unifying their treatment.

## Key findings

- Unified derivation of stable and extreme value distributions
- Highlights the universality of limit behaviors
- Provides a transparent, non-rigorous analytical approach

## Abstract

The Generalized Central Limit Theorem is a remarkable generalization of the Central Limit Theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge under appropriate scaling to a distribution belonging to a special family known as Levy stable distributions. Similarly, the maximum of i.i.d. variables may converge to a distribution belonging to one of three universality classes (Gumbel, Weibull and Frechet). Here, we rederive these known results following a mathematically non-rigorous yet highly transparent renormalization-group-like approach that captures both of these universal results following a nearly identical procedure.