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Correlation function inadequacy in random-sequence entropy measures.

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Summary
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This study links correlation functions and conditional entropies for random sequences using additive Markov chains. It reveals that approximating entropy with two-point distributions is accurate only for specific sequences.

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Area of Science:

  • Information Theory
  • Statistical Mechanics
  • Stochastic Processes

Background:

  • Understanding the relationship between statistical properties and information-theoretic measures in random sequences is crucial.
  • Additive Markov chains provide a framework for modeling sequential data with dependencies.

Purpose of the Study:

  • To establish a connection between correlation functions and conditional entropies for symbolic and numerical random sequences.
  • To evaluate the accuracy of using two-point probability distributions for entropy approximation in random chains.

Main Methods:

  • Utilizing the additive Markov chain approach.
  • Expressing entropy via two-point probability distribution functions.
  • Relating conditional entropy to correlation functions and Kullback-Leibler mutual information.

Main Results:

  • A relationship between correlation functions and conditional entropies was established for additive Markov chains.
  • Entropy approximation using two-point distributions is satisfactory only for specific random sequences.
  • The conditional entropy derived via two-point distributions is generally lower than the actual value for numerical sequences.

Conclusions:

  • The study highlights the limitations of the two-point distribution approximation for entropy estimation in general random sequences.
  • Conditional entropy for additive Markov chains can be decomposed into Kullback-Leibler mutual information.
  • An example demonstrates sequences with zero correlation functions but non-zero correlations.