Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 125Fig. 4.7 Schematic connections between various probabilistic models. From [245].4.6.2 The TMNT Model and Its Numerical ApproachHere we follow [240]. This model, in contrast with the MTG one, concerns continuousrandom variables. Let us consider N correlated uniform random variablesf (x) ={1 if − 1/2 ≤ x ≤ 1/2 ,0 otherwise.(4.54)The correlation is introduced through the following multivariate Gaussian N × Ncovariance matrix, using probability integral transform (component by component):⎛⎜⎝1 ρ ρ ... ρρ 1 ρ ... ρρ ρ 1 ... ρ... ... ... ... ...ρ ρ ρ ... 1⎞⎟⎠(4.55)with −1 ≤ ρ ≤ 1(ρ = 0 means independence; ρ = 1 means full correlation). SeeFig. 4.8 for the influence of ρ for fixed N, and Figs. 4.9 and 4.10 for the influence ofN for fixed ρ. TheN →∞limiting distribution of the sum of N random variablesappears to be very well fitted by q-Gaussians withq(ρ, N) = q ∞ (ρ) − A(ρ) . (4.56)Nδ(ρ)For example, q ∞ (0.5) ≃ 0.3545, A(0.5) ≃ 0.5338, and δ(0.5) ≃ 1.9535. Wepresent q ∞ (ρ) in Fig. 4.11. It is well fitted by the heuristic relation