相关随机变量标度不变概率模型的极限分布--以q-Gaussian为例
Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
摘要：
在适当的范数和宽度约束下的玻尔兹曼-吉布斯（BG）熵的简化产生高斯分布。此外，标准Fokker-Planck（FP）方程（与具有加性噪声的朗之万方程相关）的基本解以及中心极限定理吸引子都是高斯型的。具有这些特征的最简单的随机模型是N到无穷大的独立二进制随机变量，正如de Moivre和Laplace首先证明的那样。强相关随机变量会发生什么？这种相关性通常存在于物理情况下，例如具有远程交互或记忆的系统。q-高斯经常被观察到。如果朗之万方程包括乘性噪声，或者FP方程是非线性的，则通常是这样。标度不变性，即可交换的二进制随机过程，允许相关性和非高斯分布之间的关系的系统分析。特别是，一个对所有q（包括q>1）产生q-高斯分布的广义随机模型缺失了。这是通过使用拉普拉斯-德菲内蒂表示定理来实现的，该定理体现了可互换随机变量的严格标度不变性。我们证明了严格的标度不变性与q-高斯性一起要求相关的广泛熵是BG。
英文摘要：
Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and width constraints yields the Gaussian distribution. Also, the basic solutions of the standard Fokker-Planck (FP) equation (related to the Langevin equation with additive noise), as well as the Central Limit Theorem attractors, are Gaussians. The simplest stochastic model with such features is N to infinity independent binary random variables, as first proved by de Moivre and Laplace. What happens for strongly correlated random variables? Such correlations are often present in physical situations as e.g. systems with long range interactions or memory. Frequently q-Gaussians become observed. This is typically so if the Langevin equation includes multiplicative noise, or the FP equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic processes, allow a systematical analysis of the relation between correlations and non-Gaussian distributions. In particular, a generalized stochastic model yielding q-Gaussians for all q (including q>1) was missing. This is achieved here by using the Laplace-de Finetti representation theorem, which embodies strict scale-invariance of interchangeable random variables. We demonstrate that strict scale invariance together with q-Gaussianity mandates the associated extensive entropy to be BG.
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