Ben-Naim E, Krapivsky P L, Lemons N W
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Dec;92(6):062139. doi: 10.1103/PhysRevE.92.062139. Epub 2015 Dec 22.
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.
我们研究多个随机变量序列的极值统计。对于每个具有(N)个变量的序列，这些变量独立地从相同分布中抽取，运行最大值定义为截至目前的最大变量。我们比较(m)个独立序列的运行最大值，并研究最大值完全有序的概率(S(N))，即第一个序列的运行最大值总是大于第二个序列的运行最大值，并依次类推，第二个序列的运行最大值总是大于第三个序列的运行最大值，依此类推。概率(S(N))是通用的：它不依赖于从中抽取随机变量的分布。对于两个序列，(S(N) \sim N^{(-1/2)})，一般来说，对于大的(N)，衰减是代数形式的，(S(N) \sim N^{(-\sigma(m))})。我们通过解析得到指数(\sigma(3) \cong 1.302931)，它是一个超越方程的根。此外，指数(\sigma(m))随(m)增长，并且我们表明对于大的(m)，(\sigma(m) \sim m)。
