Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 127Fig. 4.9 TMNT model: Distribution of the sum of N = 100 random variables with ρ = 0.2. It isremarkably well fitted by a q-Gaussian with q = 0.8347 (continuous curve).q ∞ (ρ) = 1 − (5/3)ρ1 − ρ. (4.57)Let us now apply the present numerical approach to a model which generalizesthat of matrix (4.55). We assume the following covariance matrix:⎛⎞1 ρ(2) ρ(3) ... ρ(N)ρ(2) 1 ρ(2) ... ρ(N − 1)⎜ ρ(3) ρ(2) 1 ... ρ(N − 2)⎟(4.58)⎝ ... ... ... ... ... ⎠ρ(N) ρ(N − 1) ρ(N − 2) ... 1withρ(r) = ρ r α (−1 ≤ ρ ≤ 1; α ≥ 0; r = 2, 3, 4,...,N) . (4.59)As in the α = 0 case (i.e., matrix (4.55)), q-Gaussians provide an excellent fitting.The dependence of q on (ρ,α) is depicted in Fig. 4.12. This numerical result istotally consistent with what is expected in terms of the motivations of nonextensive