# Super Generalized Central Limit Theorem: Limit distributions for sums of non-identical random variables with power-laws

## TL;DR

This paper proves that sums of non-identical power-law distributed variables converge to a unique stable distribution, explaining the universality of stable laws in natural and social phenomena.

## Contribution

It establishes a super generalized central limit theorem for non-identical power-law variables, extending classical results to more complex, real-world data.

## Key findings

- Sums of non-identical power-law variables converge to a stable distribution.
- The theorem explains the universality of stable laws in diverse systems.
- Application to stock market returns demonstrates practical relevance.

## Abstract

In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the mathematical characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical stochastic processes with power-laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws such that the sums of the logarithmic return of non-identical stock price fluctuations follow stable distributions.