Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 119distribution of the sum approaches, after appropriate centering and rescaling, a Lévydistribution when N →∞.Third, we have the class (from now on referred to as the q-Gaussian class) correspondingto γ = 2 and q ≠ 1. Its basic solutions are, as already discussed,q-Gaussians. And these solutions are stable in the sense that, if we start with an arbitrary(symmetric) solution p(x, 0), it asymptotically approaches a q-Gaussian. Thishas been numerically verified and analytically proved in [272–274]. As we shall see,a generalized Central Limit Theorem (noted q-G-CLT or just q-CLT) can be establishedfor this situation. It corresponds to the violation of the hypothesis of independence.Not a weak violation with correlations that gradually disappear in the N →∞ limit, but a certain class of global correlations which persist up to infinity. Thismakes sense since Eq. (4.32) is nonlinear for q ≠ 1. The possible existence of sucha theorem was first suggested in [826], specifically conjectured in [191] and finallyproved in [247]. This theorem also demands the finiteness of a certain q-variance.If this q-variance diverges, then we are led to the fourth and last present class.The fourth class (from now on referred to as the (q,α) class) corresponds toγ ≡ α