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Title: Maxima of two random walks: Universal statistics of lead changes

In this study, we investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $${\pi }^{-1}\mathrm{ln}t$$ in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as $${t}^{-1/4}{(\mathrm{ln}t)}^{n}$$ for Brownian motion and as $${t}^{-\beta (\mu )}{(\mathrm{ln}t)}^{n}$$ for symmetric Lévy flights with index μ. The decay exponent $$\beta \equiv \beta (\mu )$$ varies continuously with the Lévy index when $$0\lt \mu \lt 2$$, and remains constant $$\beta =1/4$$ for $$\mu \gt 2$$.
Authors:
 [1] ;  [2] ;  [3]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Boston Univ., Boston, MA (United States); Univ. Paris-Saclay and CNRS, Gif-sur-Yvette (France)
  3. Univ. Paris-1 Pantheon-Sorbonne, Paris (France)
Publication Date:
OSTI Identifier:
1255157
Report Number(s):
LA-UR--15-29655
Journal ID: ISSN 1751-8113
Grant/Contract Number:
AC52-06NA25396
Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 49; Journal Issue: 20; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Research Org:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING