Ben-Naim E, Krapivsky P L
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Aug;88(2):022145. doi: 10.1103/PhysRevE.88.022145. Epub 2013 Aug 28.
We study statistics of records in a sequence of random variables. These identical and independently distributed variables are drawn from the parent distribution ρ. The running record equals the maximum of all elements in the sequence up to a given point. We define a superior sequence as one where all running records are above the average record expected for the parent distribution ρ. We find that the fraction of superior sequences S(N) decays algebraically with sequence length N, S(N)N(-β) in the limit N→∞. Interestingly, the decay exponent β is nontrivial, being the root of an integral equation. For example, when ρ is a uniform distribution with compact support, we find β=0.450265. In general, the tail of the parent distribution governs the exponent β. We also consider the dual problem of inferior sequences, where all records are below average, and find that the fraction of inferior sequences I(N) decays algebraically, albeit with a different decay exponent, I(N)N(-α). We use the above statistical measures to analyze earthquake data.
我们研究随机变量序列中记录的统计特性。这些独立同分布的变量取自母分布ρ。运行记录等于序列中截至某一给定点的所有元素的最大值。我们将一个优越序列定义为这样一种序列，其中所有运行记录都高于母分布ρ预期的平均记录。我们发现，优越序列的比例S(N)随着序列长度N呈代数衰减，在N→∞的极限情况下，S(N)N^(-β)。有趣的是，衰减指数β并非平凡的，它是一个积分方程的根。例如，当ρ是具有紧致支撑的均匀分布时，我们发现β = 0.450265。一般来说，母分布的尾部决定指数β。我们还考虑了劣质序列的对偶问题，即所有记录都低于平均值的情况，并发现劣质序列的比例I(N)也呈代数衰减，尽管衰减指数不同，I(N)N^(-α)。我们使用上述统计量来分析地震数据。
