以{PDMATHAAB q}-高斯分布为$N\to \infty$极限分布的$N$相关二元随机变量的严格渐近标度不变概率模型
Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions
摘要：
为了从物理上揭示{ \it $q$ --independence}和{ \it scale-invariance}之间的关系，我们引入了三种类型的具有二进制随机变量的渐近标度不变概率模型，即（i）族，其特征在于一个指数$ν=1,2,3,...$，统一了Leibnitz三角形（$ν=1$）和独立变量的情况（$ν\to\infty$）;（ii）$q$的两个稍微不同的离散化-高斯;（iii）一个特殊的家庭，其特征在于参数$χ$，它概括了通常情况下的独立变量（恢复为$χ=1/2$）。模型（i）和（iii）实际上是严格尺度不变的。对于模型（i），我们分析表明，$N \to\infty$概率分布是一个$q$ -高斯与$q=(ν-2)/(ν-1)$。模型（ii）的方法$q$-高斯的建设，我们数值表明，他们这样做的渐近尺度不变性。模型（iii），像最近由Hilhorst和Schehr（2007）讨论的另外两个严格尺度不变的模型一样，而是接近{ \it not} $q$-高斯分布的极限分布。出现的情况是，渐近（甚至严格）标度不变性是不够的，但它可能是必要的严格（或渐近）$q$ -独立性，这反过来又要求$q$ -高斯吸引子。
英文摘要：
In order to physically enlighten the relationship between {\it $q$--independence} and {\it scale-invariance}, we introduce three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index $ν=1,2,3,...$, unifying the Leibnitz triangle ($ν=1$) and the case of independent variables ($ν\to\infty$); (ii) two slightly different discretizations of $q$--Gaussians; (iii) a special family, characterized by the parameter $χ$, which generalizes the usual case of independent variables (recovered for $χ=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the $N \to\infty$ probability distribution is a $q$--Gaussian with $q=(ν-2)/(ν-1)$. Models (ii) approach $q$--Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are {\it not} $q$--Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) $q$--independence, which, in turn, mandates $q$--Gaussian attractors.
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