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

Max Jauregui, Constantino Tsallis
2015 Journal of Mathematical Physics  
We consider correlated random variables X_1,...,X_n taking values in {0,1} such that, for any permutation π of {1,...,n}, the random vectors (X_1,...,X_n) and (X_π(1),...,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→∞ implies independence). Putting S_n=X_1+...+X_n, the distribution of S_n-n/2 approaches a Q-Gaussian distribution with
more » ... ct 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 a beta distribution with both parameters equal to ν. In particular, the law of large numbers does not hold since, if 0< x<1/2, then P(S_n/n< x), which is the probability of the event {S_n/n< x} (large deviation), does not converges to zero as n→∞. For x=0 and every real ν>0, we show that 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 00 is an integer, we show that we can analytically find upper and lower bounds for the difference between P(S_n/n< x) and its (n→∞) 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.
doi:10.1063/1.4908108 fatcat:5y6gnmlrpnejnimvbdx4qtj5im