Scaling exponents for ordered maxima
Abstract
We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability S_{N} that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability S_{N} is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, S_{N}~N^{–1/2}, and in general, the decay is algebraic, S_{N}~N^{–σm}, for large N. We analytically obtain the exponent σ_{3}≅1.302931 as root of a transcendental equation. Moreover, the exponents σ_{m} grow with m, and we show that σ_{m}~m for large m.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Boston Univ., Boston, MA (United States); Univ. ParisSaclay, GifsurYvette (France)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1239571
 Alternate Identifier(s):
 OSTI ID: 1234086
 Report Number(s):
 LAUR1528155
Journal ID: ISSN 15393755; PLEEE8
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
 Additional Journal Information:
 Journal Volume: 92; Journal Issue: 6; Journal ID: ISSN 15393755
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; extremevalue statistics
Citation Formats
BenNaim, E., Krapivsky, P. L., and Lemons, N. W. Scaling exponents for ordered maxima. United States: N. p., 2015.
Web. https://doi.org/10.1103/PhysRevE.92.062139.
BenNaim, E., Krapivsky, P. L., & Lemons, N. W. Scaling exponents for ordered maxima. United States. https://doi.org/10.1103/PhysRevE.92.062139
BenNaim, E., Krapivsky, P. L., and Lemons, N. W. Tue .
"Scaling exponents for ordered maxima". United States. https://doi.org/10.1103/PhysRevE.92.062139. https://www.osti.gov/servlets/purl/1239571.
@article{osti_1239571,
title = {Scaling exponents for ordered maxima},
author = {BenNaim, E. and Krapivsky, P. L. and Lemons, N. W.},
abstractNote = {We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability SN that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability SN is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, SN~N–1/2, and in general, the decay is algebraic, SN~N–σm, for large N. We analytically obtain the exponent σ3≅1.302931 as root of a transcendental equation. Moreover, the exponents σm grow with m, and we show that σm~m for large m.},
doi = {10.1103/PhysRevE.92.062139},
journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics},
number = 6,
volume = 92,
place = {United States},
year = {2015},
month = {12}
}
Web of Science
Works referenced in this record:
Role of global warming on the statistics of recordbreaking temperatures
journal, December 2006
 Redner, S.; Petersen, Mark R.
 Physical Review E, Vol. 74, Issue 6
Statistical properties of recordbreaking temperatures
journal, December 2010
 Newman, William I.; Malamud, Bruce D.; Turcotte, Donald L.
 Physical Review E, Vol. 82, Issue 6
Record occurrence and record values in daily and monthly temperatures
journal, February 2013
 Wergen, G.; Hense, A.; Krug, J.
 Climate Dynamics, Vol. 42, Issue 56
Frequency of recordbreaking floods in the United States
journal, June 2001
 Vogel, Richard M.; ZafirakouKoulouris, Antigoni; Matalas, Nicholas C.
 Water Resources Research, Vol. 37, Issue 6
Emergence of patterns in random processes. II. Stochastic structure in random events
journal, June 2014
 Newman, William I.
 Physical Review E, Vol. 89, Issue 6
Recordbreaking avalanches in driven threshold systems
journal, May 2013
 Shcherbakov, Robert; Davidsen, Jörn; Tiampo, Kristy F.
 Physical Review E, Vol. 87, Issue 5
Detecting human influence on extreme temperatures in China: HUMAN INFLUENCE ON EXTREME TEMPERATURES
journal, March 2013
 Wen, Qiuzi Han; Zhang, Xuebin; Xu, Ying
 Geophysical Research Letters, Vol. 40, Issue 6
Record statistics for biased random walks, with an application to financial data
journal, May 2011
 Wergen, Gregor; Bogner, Miro; Krug, Joachim
 Physical Review E, Vol. 83, Issue 5
Nontrivial Exponent for Simple Diffusion
journal, September 1996
 Majumdar, Satya N.; Sire, Clément; Bray, Alan J.
 Physical Review Letters, Vol. 77, Issue 14
Persistent Spins in the Linear Diffusion Approximation of Phase Ordering and Zeros of Stationary Gaussian Processes
journal, September 1996
 Derrida, Bernard; Hakim, Vincent; Zeitak, Reuven
 Physical Review Letters, Vol. 77, Issue 14
Persistence and firstpassage properties in nonequilibrium systems
journal, June 2013
 Bray, Alan J.; Majumdar, Satya N.; Schehr, Grégory
 Advances in Physics, Vol. 62, Issue 3
Statistics of superior records
journal, August 2013
 BenNaim, E.; Krapivsky, P. L.
 Physical Review E, Vol. 88, Issue 2
Scaling exponent for incremental records
journal, October 2013
 Miller, P. W.; BenNaim, E.
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2013, Issue 10
Persistence of random walk records
journal, June 2014
 BenNaim, E.; Krapivsky, P. L.
 Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 25
Slow Kinetics of Brownian Maxima
journal, July 2014
 BenNaim, E.; Krapivsky, P. L.
 Physical Review Letters, Vol. 113, Issue 3
From Markovian to nonMarkovian persistence exponents
journal, February 2015
 RandonFurling, J.
 EPL (Europhysics Letters), Vol. 109, Issue 4
On the fluctuations of sums of random variables
journal, December 1953
 Andersen, Erik Sparre
 MATHEMATICA SCANDINAVICA, Vol. 1
Universal firstpassage properties of discretetime random walks and Lévy flights on a line: Statistics of the global maximum and records
journal, October 2010
 Majumdar, Satya N.
 Physica A: Statistical Mechanics and its Applications, Vol. 389, Issue 20
Walks, walls, wetting, and melting
journal, March 1984
 Fisher, Michael E.
 Journal of Statistical Physics, Vol. 34, Issue 56
Commensurate melting, domain walls, and dislocations
journal, January 1984
 Huse, David A.; Fisher, Michael E.
 Physical Review B, Vol. 29, Issue 1
Brownian motion in a Weyl chamber, noncolliding particles, and random matrices
journal, April 1999
 Grabiner, D.
 Annales de l'Institut Henri Poincare (B) Probability and Statistics, Vol. 35, Issue 2
Kinetics of a diffusive capture process: lamb besieged by a pride of lions
journal, September 1996
 Krapivsky, P. L.; Redner, S.
 Journal of Physics A: Mathematical and General, Vol. 29, Issue 17
Ordering of random walks: the leader and the laggard
journal, February 2003
 benAvraham, D.; Johnson, B. M.; Monaco, C. A.
 Journal of Physics A: Mathematical and General, Vol. 36, Issue 7
Kinetics of first passage in a cone
journal, November 2010
 BenNaim, E.; Krapivsky, P. L.
 Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 49
Firstpassage exponents of multiple random walks
journal, November 2010
 BenNaim, E.; Krapivsky, P. L.
 Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 49
Density relaxation in a vibrated granular material
journal, May 1995
 Knight, James B.; Fandrich, Christopher G.; Lau, Chun Ning
 Physical Review E, Vol. 51, Issue 5
Random Packings of Frictionless Particles
journal, January 2002
 O'Hern, Corey S.; Langer, Stephen A.; Liu, Andrea J.
 Physical Review Letters, Vol. 88, Issue 7
Geometrical structure of disordered sphere packings
journal, June 2005
 Aste, T.; Saadatfar, M.; Senden, T.
 Physical Review E, Vol. 71, Issue 6
Works referencing / citing this record:
Record statistics of a strongly correlated time series: random walks and Lévy flights
journal, July 2017
 Godrèche, Claude; Majumdar, Satya N.; Schehr, Grégory
 Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 33