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Binary N-step Markov chains and long-range correlated systems.

O V Usatenko1, V A Yampol'skii

  • 1A. Ya. Usikov Institute for Radiophysics and Electronics, Ukrainian Academy of Science, 12 Proskura Street, 61085 Kharkov, Ukraine.

Physical Review Letters
|April 12, 2003
PubMed
Summary
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This study introduces a new theory for systems with long-range correlations using N-step Markov chains. The research reveals nonlinear variance dependence on length and self-similarity in stochastic processes.

Area of Science:

  • Stochastic processes
  • Information theory
  • Statistical mechanics

Background:

  • Systems with long-range correlations exhibit complex behaviors.
  • Understanding these correlations is crucial in various scientific fields.
  • Previous models often simplify or overlook intricate dependencies.

Purpose of the Study:

  • To develop a novel theory for systems with long-range correlations.
  • To model systems using binary N-step Markov chains.
  • To analyze the statistical properties of these systems.

Main Methods:

  • Development of a theoretical framework for binary N-step Markov chains.
  • Analytical and numerical computation of correlation and distribution functions.
  • Investigation of variance and its dependence on word length (L).

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Main Results:

  • The conditional probability is a linear function of preceding symbols.
  • Variance shows a nonlinear dependence on length L for non-extreme correlations.
  • A self-similarity characteristic of the stochastic process was identified.

Conclusions:

  • The developed theory provides a robust model for systems with long-range correlations.
  • The findings are applicable to analyzing coarse-grained written and DNA texts.
  • The study highlights the nonlinear dynamics and self-similarity in complex systems.