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(-σ(m)), for large N. We analytically obtain the exponent σ(3)≅1.302931 as root of a transcendental equation. Furthermore, the exponents σ(m) grow with m, and we show that σ(m)∼m for large m.