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We consider correlated random variables $X_1,\dots,X_n$ taking values in$\{0,1\}$ such that, for any permutation $\pi$ of $\{1,\dots,n\}$, the randomvectors $(X_1,\dots,X_n)$ and $(X_{\pi(1)},\dots,X_{\pi(n)})$ have the samedistribution. This distribution, which was introduced by Rodr\'iguez et al(2008) and then generalized by Hanel et al (2009), is scale-invariant anddepends on a real parameter $\nu>0$ ($\nu\to\infty$ 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 ($Q=1-1/(\nu-1)<1$) as $n$increases, after appropriate scaling. In the present article, we show that thedistribution of $S_n/n$ converges, as $n\to\infty$, to a beta distribution withboth parameters equal to $\nu$. In particular, the law of large numbers doesnot hold since, if $0\le x<1/2$, then $\mathbb{P}(S_n/n\le x)$, which is theprobability of the event $\{S_n/n\le x\}$ (large deviation), does not convergesto zero as $n\to\infty$. For $x=0$ and every real $\nu>0$, we show that$\mathbb{P}(S_n=0)$ decays to zero like a power law of the form $1/n^\nu$ witha subdominant term of the form $1/n^{\nu+1}$. If $0<x\le 1$ and $\nu>0$ is aninteger, we show that we can analytically find upper and lower bounds for thedifference between $\mathbb{P}(S_n/n\le x)$ and its ($n\to\infty$) limit. Wealso show that these bounds vanish like a power law of the form $1/n$ with asubdominant term of the form $1/n^2$.

journal of mathematical physics

Convergence of the probability of large deviations in a model of  correlated random variables having compact-support $Q$-Gaussians as limiting  distributions