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Distribution of returns and its asymptotic behavior.

Silvia A Menchón1, P Román2

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This study analyzes return distributions in one-dimensional random walks, revealing a universal power-law decay in symmetric cases. Deviations from symmetry introduce exponential corrections, linking to nonadditive statistical mechanics.

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Area of Science:

  • Statistical Mechanics
  • Complex Systems
  • Probability Theory

Background:

  • Self-organized criticality and random walk theory are key frameworks for understanding complex system dynamics.
  • Distributions of returns in financial markets and physical processes often exhibit heavy tails, deviating from normal distributions.

Purpose of the Study:

  • To investigate the distributions of returns for one-dimensional random walks with nearest-neighbor jumps.
  • To characterize the asymptotic behavior of these distributions and their connection to nonadditive statistical mechanics.

Main Methods:

  • Analysis of discrete-time, discrete-space one-dimensional random walks.
  • Derivation of explicit expressions for return distributions for specific parameter sets.
  • Development of an integral representation and rigorous characterization of asymptotic behavior.

Main Results:

  • A universal asymptotic power-law decay (exponent -3/2) was identified for symmetric random walks.
  • Deviations from symmetry were shown to introduce exponential corrections to the return distribution.
  • A q-Gaussian fit implies a nonadditive parameter q=7/3, connecting the model to nonadditive statistical mechanics.

Conclusions:

  • The study provides a framework for understanding fat-tailed return distributions in discrete random walk models.
  • The findings link random walk dynamics to concepts in nonadditive statistical mechanics.
  • Explicit mathematical expressions and asymptotic behaviors of return distributions are established.