Anomalous scaling due to correlations: Limit theorems and self-similar processes

TL;DR

This paper establishes limit theorems explaining how correlated random variables can exhibit persistent anomalous scaling in their sum's probability density, with implications for critical phenomena and non-Markovian processes.

Contribution

It introduces explicit mechanisms and limit theorems for anomalous scaling in sums of correlated variables, highlighting universality and stability properties.

Findings

Derives theorems characterizing anomalous scaling in correlated sums.

Justifies the universal nature of anomalous scaling forms.

Provides models of non-Markovian processes with anomalous scaling.

Abstract

We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling forms, justify their universal character, and specify universality domains in the spaces of joint probability density functions of the summand variables. These density functions are assumed to be invariant under arbitrary permutations of their arguments. Examples from the theory of critical phenomena are discussed. The novel notion of stability implied by the limit theorems also allows us to define sequences of random variables whose sum satisfies anomalous scaling for any finite number of summands. If regarded as developing in time, the stochastic processes described by these variables are non-Markovian generalizations of Gaussian processes with uncorrelated increments, and provide, e.g., explicit realizations of a recently proposed model of index evolution in finance.