经济学中的幂律分布：非广延统计方法
Power-law distributions in economics: a nonextensive statistical approach
摘要：
玻尔兹曼-吉布斯（$BG$）统计力学的基石是玻尔兹曼-吉布斯-杰恩斯-香农熵$S_{BG} \equiv -k\int dx f(x)\ln f(x)$，其中$k$是正常数，$f(x)$是概率密度函数。这一理论在一个多世纪以来，在处理时空短相关性占主导地位的系统方面取得了巨大的成功。然而，有一些反常的自然和人工系统违反了其适用性的基本要求。不同的物理熵，而不是标准的，似乎是必要的，以令人满意地处理这种异常。其中一个熵是$S_q \equiv k (1-\int dx [f(x)]^q)/(1-q)$（具有$S_1=S_{BG}$），其中熵指数$q$是实参数。它已被提议作为$BG$理论的推广的基础，称为{ \it非广延统计力学}。$S_q$与$S_{BG}$共有四个显著的性质，即{ PDMATHABMT_n}（$\forall q>0$）、{ PDMATHABMT_Lesche-稳定性}（$\forall q>0$）、{ PDMATHABMT_n单位时间熵产生的有限性}（$q \in \Re$）和{ PDMATHABMT_n可加性}（对于至少包括$q=1$的$q$的紧凑支持）。这些性质的同时有效性表明，$S_q$是适当的桥接，在宏观层面上，与经典热力学本身。指数概率函数出现在标准环境中的自然方式与此相同，幂律尾分布，即使指数{PDMATHAMT out}在Lévy范围内，也出现在非广延框架中。在这篇评论中，我们打算表明，许多感兴趣的经济过程中，厚尾概率函数的经验观察，可以描述的统计机制，基础上的非广延理论。
英文摘要：
The cornerstone of Boltzmann-Gibbs ($BG$) statistical mechanics is the Boltzmann-Gibbs-Jaynes-Shannon entropy $S_{BG} \equiv -k\int dx f(x)\ln f(x)$, where $k$ is a positive constant and $f(x)$ a probability density function. This theory has exibited, along more than one century, great success in the treatment of systems where short spatio/temporal correlations dominate. There are, however, anomalous natural and artificial systems that violate the basic requirements for its applicability. Different physical entropies, other than the standard one, appear to be necessary in order to satisfactorily deal with such anomalies. One of such entropies is $S_q \equiv k (1-\int dx [f(x)]^q)/(1-q)$ (with $S_1=S_{BG}$), where the entropic index $q$ is a real parameter. It has been proposed as the basis for a generalization, referred to as {\it nonextensive statistical mechanics}, of the $BG$ theory. $S_q$ shares with $S_{BG}$ four remarkable properties, namely {\it concavity} ($\forall q>0$), {\it Lesche-stability} ($\forall q>0$), {\it finiteness of the entropy production per unit time} ($q \in \Re$), and {\it additivity} (for at least a compact support of $q$ including $q=1$). The simultaneous validity of these properties suggests that $S_q$ is appropriate for bridging, at a macroscopic level, with classical thermodynamics itself. In the same natural way that exponential probability functions arise in the standard context,power-law tailed distributions, even with exponents {\it out} of the Lévy range, arise in the nonextensive framework. In this review, we intend to show that many processes of interest in economy, for which fat-tailed probability functions are empirically observed, can be described in terms of the statistical mechanisms that underly the nonextensive theory.
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