A formalism for studying long-range correlations in many-alphabets sequences

TL;DR

This paper develops a mean-field theory for long-range correlations in sequences with multiple alphabets, revealing how parameters influence variance behavior and showing that coarse-graining can alter perceived correlations.

Contribution

It introduces a formalism for analyzing long-range correlations in multi-alphabet sequences and provides exact solutions for specific cases, highlighting the effects of coarse-graining.

Findings

Variance can be linear or superlinear depending on parameters.

Exact solutions for three- and four-alphabet sequences are provided.

Coarse-graining may change the apparent long-range correlations.

Abstract

We formulate a mean-field-like theory of long-range correlated $L$-alphabets sequences, which are actually systems with $(L-1)$ independent parameters. Depending on the values of these parameters, the variance on the average number of any given symbol in the sequence shows a linear or a superlinear dependence on the total length of the sequence. We present exact solution to the four-alphabets and three-alphabets sequences. We also demonstrate that a mapping of the given sequence into a smaller alphabets sequence (namely, a {\it coarse-graining} process) does not necessarily imply that long-range correlations found in the latter would correspond to those of the former.