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Scaling exponents for ordered maxima.

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Summary
This summary is machine-generated.

This study explores extreme value statistics in random variable sequences. The probability of perfectly ordered running maxima across multiple sequences is universal and decays algebraically with sequence length.

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Area of Science:

  • Probability theory
  • Extreme value statistics
  • Stochastic processes

Background:

  • Understanding the behavior of random variables in sequences is crucial in various scientific fields.
  • Extreme value statistics focuses on the properties of maximum or minimum values within a dataset.
  • The concept of running maxima tracks the peak value observed up to a certain point in a sequence.

Purpose of the Study:

  • To investigate the probability of perfectly ordered running maxima across multiple independent sequences of random variables.
  • To determine the universality and scaling behavior of this ordering probability.
  • To analytically derive the decay exponent for a specific number of sequences and analyze its trend with increasing sequence count.

Main Methods:

  • Analysis of extreme value statistics for multiple sequences of random variables.
  • Definition and comparison of running maxima across 'm' independent sequences.
  • Derivation of the probability S(N) for perfectly ordered running maxima.
  • Analytical calculation of the decay exponent σ(m) using transcendental equations.

Main Results:

  • The probability S(N) of perfectly ordered running maxima is universal, independent of the underlying distribution.
  • For two sequences, the decay is S(N)∼N^(-1/2).
  • In general, S(N) exhibits algebraic decay S(N)∼N^(-σ(m)) for large N.
  • The exponent σ(3) was analytically found to be approximately 1.302931.
  • The exponent σ(m) increases with 'm', showing asymptotic behavior σ(m)∼m for large 'm'.

Conclusions:

  • The ordering of running maxima in multiple random sequences follows universal statistical laws.
  • The decay rate of this ordering probability provides insights into the dependence structure of the sequences.
  • The derived exponents offer quantitative measures for the likelihood of perfect ordering as sequence length and number increase.