Scaling Exponents for Ordered Maxima

TL;DR

This paper analyzes the probability that the running maxima of multiple independent sequences of random variables are perfectly ordered, revealing universal algebraic decay and deriving growth behavior of the associated exponents.

Contribution

It introduces a universal framework for the probability of ordered maxima in multiple sequences and analytically derives key exponents, including a transcendental solution for three sequences.

Findings

Probability S_N decays as N^(-1/2) for two sequences

Explicit exponent sigma_3 ≈ 1.303 derived from transcendental equation

Exponents sigma_m grow linearly with m for large m

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^(-\sigma_m), for large N. We analytically obtain the exponent sigma_3= 1.302931 as root of a transcendental equation. Furthermore, the exponents sigma_m grow with m, and we show that sigma_m ~ m for large m.