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Universal order statistics of random walks.

Grégory Schehr¹, Satya N Majumdar

¹Université Paris-Sud, CNRS, LPT, 91405 Orsay Cedex, France. gregory.schehr@th.u-psud.fr

Physical Review Letters

|March 10, 2012

Summary

We analyzed order statistics for random walks, finding stationary gap distributions between time series maxima. This universal behavior shows algebraic decay and a power-law tail in the gap probability density.

Area of Science:

Probability Theory
Statistical Physics
Time Series Analysis

Background:

Random walks are fundamental models in statistical physics.
Understanding the behavior of time series maxima is crucial in various fields.
Order statistics provide insights into the distribution of extreme values.

Purpose of the Study:

To analytically investigate the order statistics of a symmetric random walk.
To characterize the stationary behavior of gaps between consecutive maxima in a time series.
To uncover universal properties of these gap statistics.

Main Methods:

Analytical study of order statistics for a symmetric random walk with finite variance steps.
Derivation of the stationary distribution for the gap between the k-th and (k+1)-th maximum.

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Analysis of the probability density function (pdf) and its scaling properties.

Main Results:

The gap statistics between time series maxima become stationary as the number of steps (n) approaches infinity.
The mean stationary gap exhibits a universal algebraic decay for large k, ~1/sqrt(2πk).
The stationary gap pdf shows scaling behavior with a universal function P(x) possessing a power-law tail P(x) ~ x^(-4).

Conclusions:

The gap statistics of random walk time series exhibit rich, universal behavior independent of jump distribution details.
The discovered scaling and power-law tail in the gap pdf are significant findings.
The study reveals unusual multiscaling behavior in the moments of the gap, highlighting complex statistical properties.