Nonextensive Statistical Mechanics

4.4 Correlated Anomalous Diffusion 1130.50.4(a)q → –∞q = –5q = 1p(x)0.30.2q = 2q = 2.50.1q → 300.50.4–4 –20 2 4x(b)p(0)0.30.20.10–3 –2 –1 0q1 2 3Fig. 4.1 (a) Distributions p q (x)/ √ β vs. √ β x for typical values of q, [β = 1/(3 − q)] (b) Valuesof p q (0)/ √ β as a function of q.vi. If q = 3+m where m is a positive integer, these distributions recover the1+mStudent’ s t-distributions with m degrees of freedom. 2 If q = n−6n−4 where n >4 is a positive integer, these distributions recover the so-called r-distributions[271].vii. The relation q ↔ (2 − q) existing between the nonlinearity of the Fokker–Planck-like equation and its q-exponential solution is remarkable.2 In the financial literature, these q-Gaussian distributions with q > 1 emerge quite frequently(p(x) ∝ 1/(a 2 + x 2 ) η with η>0). They are referred to as Student’s distributions for any real valueof η ≡ 1/(q − 1). Strictly speaking, this is an abusive notation.