Central limit theorem for anomalous scaling due to correlations

TL;DR

This paper establishes a central limit theorem for sums of critically correlated variables, linking anomalous scaling behaviors to Lévy processes, and enabling the analysis of correlations from global scaling properties.

Contribution

It introduces a CLT for correlated variables with anomalous scaling, connecting correlated processes to Lévy diffusion and enabling correlation analysis from scaling.

Findings

Correlated anomalous diffusion can be mapped onto Lévy diffusion.

The theorem characterizes asymptotic distributions for critically correlated variables.

Correlation structures can be inferred from global anomalous scaling.

Abstract

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the L\'evy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto L\'evy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.