级数定律
The law of series
摘要：
考虑了多个态的正熵遍历过程。我们的第一个主要结果断言，第一次访问的标准化等待时间的分布函数是一个小的（即，对于大多数这样的气缸并且直到100%，气缸组$B$由指数分布函数$1-e^{-t}$支配。也就是说，沿时间轴的如此理解的“罕见事件”$B$的发生可以以接近指数分布的间隙大小出现（如在独立伯努利过程中），或者它们彼此“吸引”。我们的第二个主要结果指出，一个{ \it典型}遍历过程的正熵具有以下性质：分布函数的归一化首中时的大多数圆柱$B$的长度$n'$收敛到零沿\sq \ $n'$的上密度为1。这种圆柱体$B$的出现“强烈吸引”，即，它们出现在许多频繁重复的“系列”中，这些重复被几乎完全不存在的巨大间隙隔开。这些结果，如果得到恰当而仔细的解释，就可以从纯粹的统计学角度，独立于物理学，对一个有着百年历史（迄今为止，严肃的科学一直在回避）的常识性现象--{ \it级数定律}--提供一些新的解释。这个现象表明，现实中的罕见事件一旦发生，就有一种神秘的不适时重复的倾向。
英文摘要：
We consider an ergodic process on finitely many states, with positive entropy. Our first main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. That is, the occurrences of so understood "rare event" $B$ along the time axis can appear either with gap sizes of nearly exponential distribution (like in the independent Bernoulli process), or they "attract" each-other. Our second main result states that a {\it typical} ergodic process of positive entropy has the following property: the distribution functions of the normalized hitting times for the majority of cylinders $B$ of lengths $n'$ converge to zero along a \sq\ $n'$ whose upper density is 1. The occurrences of such a cylinder $B$ "strongly attract", i.e., they appear in "series" of many frequent repetitions separated by huge gaps of nearly complete absence. These results, when properly and carefully interpreted, shed some new light, in purely statistical terms, independently from physics, on a century old (and so far rather avoided by serious science) common-sense phenomenon known as {\it the law of series}, asserting that rare events in reality, once occurred, have a mysterious tendency for untimely repetitions.
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