在一般的金融理论里，证券价格被假定服从正态分布，著名的布-斯期权定价公式就以正态分布为基础。但实证研究发现证券价格并不服从正态分布，而是服从一种尖峰厚尾的分布，就是说，出现极端数据的概率比正态分布的概率要大，用幂次分布拟合的效果比正态分布拟合的效果好，而水文学中洪峰事件可能也是服从类似的分布。

哪幂次分布的分布函数是什么。

the following word is what I found from a essay,

previous work(Xavier et al, Nature423 2003 ) analyzed the distribution of R for the 1000 largest US stocks and several major international indexes and found a power-law distrubution P{[R]>x} ~x(-n), where n~=3, which si called the cuboc law of returns.

Power Law Tails is developed from Zipf's Law.

In an investing context, we might expect to encounter this law in some
important instances: (a) the capitalization of stocks in the S&P 500 index and
(b) the capitalization of a large number of similar traders in a market, and
perhaps (c) the distribution of sizes of trading firms. Empirical evidence has
been published supporting (a) and (b).

A related law governs distribution of log returns of securities without the
requirement of a slope of -1. For a given security, this law will take the form
P[R>r]~ 1/(r^k), for large r
where R is a random variable of asset or portfolio returns and is a constant.
Note that the probability density function (if it exists) will have a tail
~ 1/[r^(k+1)]. A probability distribution that has this property is said to have power law tails.

In distributions with power law tails, the
occurrence of large deviations is much more likely than in those with
exponentially decaying tails.

A Theory of Power Law Distributions in Financial Fluctuations,
by Xavier Gabaix, et. al., contained in his home page
http://pages.stern.nyu.edu/~xgabaix/.

 http://pages.stern.nyu.edu/~xgabaix/papers/powerLaws.pdf

"Power Laws", Entry for The New Palgrave Dictionary of Economics, 2nd Edition.

Most "laws" in economics are power laws: a survey of theory and empirics on power laws.

上面是Xavier Gabaix 为新帕尔格雷夫经济学词典写的专文。

在金融历史上学术界对市场规律的探索经历了200多年范式漂移：PARETO（1897年）首次发现经济现象中一些事件与正态分布不完全吻合，而与他称为power law 的数学规律相合，这个power law定义了一种关系，甲事件与乙事件的n次方尺度相关。因为这个相关关系的存在，使正太曲线的一尾***为肥尾巴，比如泡沫经济的增长，它出现在正太曲线的高端右侧；而经济危机的肥尾在左边。1949年Zipi发现power law与频率相关。1963年Mandelbrot重新发现power law肥尾现象于是造成了他的分形几何学问世，但是在当时经济学界的效益市场所谓正统标准模型一统天下，power law 没有引起重视。直到1987年股市崩盘，宣判了统标准模型对稀罕事件预报无能为力。当时本人在新疆石河子农学院攻读单一自由度方差分析风险的在职博士论文，因此对风险的方差量度很关注。后来到国外本人在发明《分形针灸》，就是应用频率共振的power law解释了分形针灸的止痛作用，回国后从事金融市场研究， 结合power law 与亚当斯密‘归于invision hand’ ，凯恩斯进而用一个系统比喻’股市如选美‘都与黄金分割理论相通，用来解释泡沫经济如何生成任何爆裂， 风险如何调控都能够有个说法,详见《金融正前方》2012-4