具有紧支集$Q$ -高斯分布的相关随机变量模型中大偏差概率的收敛性
Convergence of the probability of large deviations in a model of correlated random variables having compact-support $Q$-Gaussians as limiting distributions
摘要：
我们考虑相关随机变量$X_1,\dots,X_n$在$\{0,1\}$中取值，使得对于$\{1,\dots,n\}$的任何置换$π$，随机向量$(X_1,\dots,X_n)$和$(X_{π(1)},\dots,X_{π(n)})$具有相同的分布。这种分布由Rodríguez等人（2008）引入，然后由Hanel等人（2009）推广，是尺度不变的，并且取决于实参数$ν>0$（$ν\to\infty$意味着独立性）。放置$S_n=X_1+\cdots+X_n$，在适当缩放之后，随着$n$增加，$S_n-n/2$的分布接近具有紧凑支持的$Q$ -高斯分布（$Q=1-1/(ν-1)<1$）。在本文中，我们表明，$S_n/n$的分布收敛，$n\to\infty$，β分布与两个参数等于$ν$。特别地，大数定律不成立，因为如果$0\le x<1/2$，则作为事件$\{S_n/n\le x\}$（大偏差）的概率的$\mathbb{P}(S_n/n\le x)$不像$n\to\infty$那样收敛到零。对于$x=0$和每一个真正的$ν>0$，我们表明，$\mathbb{P}(S_n=0)$衰减到零的形式$1/n^ν$与次优势项的形式$1/n^{ν+1}$一样的幂律。如果$0<x\le 1$和$ν>0$是一个整数，我们表明，我们可以解析地找到$\mathbb{P}(S_n/n\le x)$和它的（$n\to\infty$）限制之间的差异的上限和下限。我们还表明，这些边界消失的形式$1/n$与次优势项的形式$1/n^2$的幂律。
英文摘要：
We consider correlated random variables $X_1,\dots,X_n$ taking values in $\{0,1\}$ such that, for any permutation $π$ of $\{1,\dots,n\}$, the random vectors $(X_1,\dots,X_n)$ and $(X_{π(1)},\dots,X_{π(n)})$ have the same distribution. This distribution, which was introduced by Rodríguez et al (2008) and then generalized by Hanel et al (2009), is scale-invariant and depends on a real parameter $ν>0$ ($ν\to\infty$ implies independence). Putting $S_n=X_1+\cdots+X_n$, the distribution of $S_n-n/2$ approaches a $Q$-Gaussian distribution with compact support ($Q=1-1/(ν-1)<1$) as $n$ increases, after appropriate scaling. In the present article, we show that the distribution of $S_n/n$ converges, as $n\to\infty$, to a beta distribution with both parameters equal to $ν$. In particular, the law of large numbers does not hold since, if $0\le x<1/2$, then $\mathbb{P}(S_n/n\le x)$, which is the probability of the event $\{S_n/n\le x\}$ (large deviation), does not converges to zero as $n\to\infty$. For $x=0$ and every real $ν>0$, we show that $\mathbb{P}(S_n=0)$ decays to zero like a power law of the form $1/n^ν$ with a subdominant term of the form $1/n^{ν+1}$. If $0<x\le 1$ and $ν>0$ is an integer, we show that we can analytically find upper and lower bounds for the difference between $\mathbb{P}(S_n/n\le x)$ and its ($n\to\infty$) limit. We also show that these bounds vanish like a power law of the form $1/n$ with a subdominant term of the form $1/n^2$.
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