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. doi: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
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