关于$q$ -Gaussian族的鲁棒性
On the robustness of the $q$-Gaussian family
摘要：
我们引入了$N$ -体概率模型的三种变形，分别称为$α$ -，$β$ -和$γ$ -变形，该模型首先由Rodríguez等人（2008）提出，具有$q$ -高斯作为$N\to\infty$极限概率分布。所提出的$α$ -和$β$ -变形是渐近尺度不变的，而$γ$ -变形不是。我们证明，对于$α$ -和$β$ -变形，所得到的变形三角形仍然有$q$ -高斯作为限制分布，$q$的值独立（依赖）的变形参数在$α$ -情况下（$β$ -情况下）。相反，在$γ$ -情况下，我们使用了著名的$Q$ -数和高斯二项式系数，产生了$q$ -高斯族之外的其他极限概率分布函数。这些结果表明，尺度不变性可能发挥重要作用的鲁棒性的$q$ -高斯家庭。
英文摘要：
We introduce three deformations, called $α$-, $β$- and $γ$-deformation respectively, of a $N$-body probabilistic model, first proposed by Rodríguez et al. (2008), having $q$-Gaussians as $N\to\infty$ limiting probability distributions. The proposed $α$- and $β$-deformations are asymptotically scale-invariant, whereas the $γ$-deformation is not. We prove that, for both $α$- and $β$-deformations, the resulting deformed triangles still have $q$-Gaussians as limiting distributions, with a value of $q$ independent (dependent) on the deformation parameter in the $α$-case ($β$-case). In contrast, the $γ$-case, where we have used the celebrated $Q$-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the $q$-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the $q$-Gaussian family.
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