Physics – Condensed Matter – Statistical Mechanics

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2009-09-04

J. Stat. Mech. (2010) P02018

Physics

Condensed Matter

Statistical Mechanics

Through text revision. 15 pages, 3 figures

Scientific paper

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.

**Baldovin Fulvio**

Physics – Condensed Matter – Statistical Mechanics

Scientist

**Stella Attilio L.**

Physics – Condensed Matter – Statistical Mechanics

Scientist

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