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

TL;DR

This paper studies the convergence properties of correlated binary random variables with scale-invariant distributions, revealing non-classical large deviation behaviors and convergence to beta distributions, challenging traditional law of large numbers assumptions.

Contribution

It demonstrates that for a specific class of correlated variables, the distribution of their normalized sum converges to a beta distribution, and large deviations do not vanish as in independent cases.

Findings

Distribution of $S_n/n$ converges to a beta distribution with parameters $ u$.

Large deviations do not tend to zero, violating the law of large numbers.

Probability of zero sum decays as a power law $1/n^ u$.

Abstract

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 $1/n^2$.