Scaling exponents for ordered maxima

Journal Article · 22 December 2015 · Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

DOI: https://doi.org/10.1103/PhysRevE.92.062139 · OSTI ID:1239571

Ben-Naim, E. ^[1]; Krapivsky, P. L. ^[2]; Lemons, N. W. ^[1]

---

1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2. Boston Univ., Boston, MA (United States); Univ. Paris-Saclay, Gif-sur-Yvette (France)

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 σ₃≅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.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC52-06NA25396
• OSTI ID: 1239571
• Alternate ID(s): OSTI ID: 1234086
• Report Number(s): LA-UR-15-28155; PLEEE8
• Journal Information: Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, Vol. 92, Issue 6; ISSN 1539-3755
• Publisher: American Physical Society (APS)Copyright Statement
• Country of Publication: United States
• Language: English