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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:
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 Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING
OSTI Identifier:
1255157

Ben-Naim, E., Krapivsky, P. L., and Randon-Furling, J.. Maxima of two random walks: Universal statistics of lead changes. United States: N. p., Web. doi:10.1088/1751-8113/49/20/205003.
Ben-Naim, E., Krapivsky, P. L., & Randon-Furling, J.. Maxima of two random walks: Universal statistics of lead changes. United States. doi:10.1088/1751-8113/49/20/205003.
Ben-Naim, E., Krapivsky, P. L., and Randon-Furling, J.. 2016. "Maxima of two random walks: Universal statistics of lead changes". United States. doi:10.1088/1751-8113/49/20/205003. https://www.osti.gov/servlets/purl/1255157.
@article{osti_1255157,
title = {Maxima of two random walks: Universal statistics of lead changes},
author = {Ben-Naim, E. and Krapivsky, P. L. and Randon-Furling, J.},
abstractNote = {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 $\mu \gt 2$.},
doi = {10.1088/1751-8113/49/20/205003},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 20,
volume = 49,
place = {United States},
year = {2016},
month = {4}
}