Nonextensive Statistical Mechanics

338 Appendix B Escort Distributions and q-Expectation Valuesis finite for all values of n (including for n →∞) only if α>3, but σ (∞) divergesfor 1 1: see Fig. B.2.As a matter of fact, the moments of all orders are finite for α>1 if, instead ofthe original distribution f n (x), we use the appropriate escort distributions [258].with q m = mq − (m − 1) andm = 0, 1, 2, 3, ..., all these moments are finite for any α>1 and any n, and theyall diverge for α ≤ 1 and n →∞(see also Section 4.7).Summarizing,Indeed, if we consider the mth order moment 〈x m 〉 (n)q m(i) If we want to characterize, for all values of n (including n →∞), the functionaldensity form (B.1) for all α>1, we can perfectly well do so by using theappropriate escort distributions, whereas the standard mean value is admissibleonly for α>2, and the standard variance is admissible only for α>3;(ii) If we only want to characterize, for all α>1 and finite n, which seismic region(in our example with earthquakes) is more dangerous, we can do so either withthe standard mean value or with the q-mean value; obviously, the larger n is,the more seismically dangerous the region is;(iii) If we only want to characterize, for all α > 1 and finite n, the size of thefluctuations, we can do so either with the standard variance or with the q-variance; obviously, the larger n is, the larger the fluctuations are.As we have illustrated, the problem of the empirical verification of a specificanalytic form for a distribution of probabilities theoretically argued is quite differentfrom the problem on how successive experimental data keep filling this functionalform. In particular, the problem of its largest empirical values constitutes an entirebranch of mathematical statistics, usually referred to as extreme value statistics (orextreme value theory) (see, for instance, [883]), and remains out of the scope of thepresent book.2 For the present purpose, we can also use 〈(x −〈x〉 q ) 2 〉 (n)2q−1 =〈x 2 〉 (n)In contrast, we cannot use 〈x 2 〉 (n)2q−1 − (〈x〉(n) q〈x〉(n) 2q−1 +(〈x〉(n)2q−1 −2〈x〉(n) q)2 ; indeed, it becomes negative for n large enough.q )2 .