  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 random
vectors $(X_1,\dots,X_n)$ and $(X_{\pi(1)},\dots,X_{\pi(n)})$ have the same
distribution. This distribution, which was introduced by Rodr\'iguez et al
(2008) and then generalized by Hanel et al (2009), is scale-invariant and
depends 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 the
distribution of $S_n/n$ converges, as $n\to\infty$, to a beta distribution with
both parameters equal to $\nu$. 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 $\nu>0$, we show that
$\mathbb{P}(S_n=0)$ decays to zero like a power law of the form $1/n^\nu$ with
a subdominant term of the form $1/n^{\nu+1}$. If $0<x\le 1$ and $\nu>0$ 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 $1/n^2$.
