Distribution of returns and its asymptotic behavior

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.