Nonextensive Statistical Mechanics

124 4 Stochastic Dynamical Foundations of Nonextensive Statistical MechanicsQ ≡− B A q , (4.50)λ ≡− ADBC , (4.51)andf λ (x) ≡1 − x1 + λ x . (4.52)atNotice that f λ (x) satisfies f λ (0) = 1 and f λ (1) = 0. It has a fixed point located⎧1⎪⎨Q = Q ∗ λ (√ 1 + λ − 1) if λ>0 ,1=if λ = 0 ,2 ⎪⎩− 1 λ (1 − √ 1 + λ) if − 1 ≤ λ 1 (in this case, one should of course avoid to scale,after centering, the variable n in such a way that it yields a compact support, as itoccurs in Figs. 4.4 and 4.5). Further related analytical and numerical results can befound in [184, 185]. These results provided a preliminary basis that reinforced theconjecture of the existence of the q − CLT.Let us remind that, in the (q,γ) plane of Fig. 4.3, we have addressed four classesof CLTs. These various CLTs will be shown to correspond to classes of globalcorrelations or absence of correlations. Two of those theorems violate the traditionalhypothesis of independence of the random variables that are being summed or arithmeticallyaveraged. In what concerns the region that simultaneously has q > 1 andγ