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

Abstract

We derive theorems which outline explicit mechanisms by which anomalous scaling for theprobability density function of the sum of many correlated random variables asymptoticallyprevails. The results characterize general anomalous scaling forms, explain their universalcharacter, and specify universality domains in the spaces of joint probability densityfunctions of the summand variables. These density functions are assumed to beinvariant under arbitrary permutations of their arguments. Examples from the theoryof critical phenomena are discussed. The novel notion of stability implied bythe limit theorems also allows us to define sequences of random variables whosesum satisfies anomalous scaling for any finite number of summands. If regardedas developing in time, the stochastic processes described by these variables arenon-Markovian generalizations of Gaussian processes with uncorrelated increments, andprovide, e.g., explicit realizations of a recently proposed model of index evolution infinance.