Nonextensive Statistical Mechanics

4.7 Central Limit Theorems 135Fig. 4.17 Q c − 1 = Q N,N/2 − 1asafunctionofN for different values of q for discretizations D1(top) andD2(bottom). The power law has exponent −2 (from [244]).4.7 Central Limit TheoremsThe standard and Lèvy–Gnedenko central limit theorems (CLT)areq-generalizedin [247–250] (see also [251–255]).We start with a definition. Theq-Fourier transform (q-FT)ofafunction f (x) isdefined as follows:∫F q [ f ](ξ) ≡ dx eq i ξ x ⊗ q f (x) . (4.70)This definition holds for any real value of q. However, its implementation is verysimple for q ≥ 1. We shall therefore restrict to this interval from now on. 6 It can beshown [247] that6 The q-Fourier transform for q < 1 can be conveniently handled, as recently shown [K.P. Nelsonand S. Umarov, The relationship between Tsallis statistics, the Fourier transform, and nonlinearcoupling, 0811.3777[cs.IT]], by using the self-dual transformation q ↔ (5 − 3q)/(3 − q), whichtransforms the q ≤ 1 interval into the 1 ≤ q < 3 interval and reciprocally.