q-generalization of Symmetric Alpha-Stable Distributions: Part II

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1 -Generalization of Symmetric Alpha-Stable Distributions: Part II Sabir Umarov Constantino Tsallis Murray Gell-Mann Stanly Steinberg SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 -GENERALIZATION OF SYMMETRIC -STABLE DISTRIBUTIONS. PART II Sabir Umarov,2, Constantino Tsallis 3,4, Murray Gell-Mann 3 and Stanly Steinberg Department of Mathematics and Statistics University of New Mexico, Albuuerue, NM 873, USA 2 Department of Mathematical Physics National University of Uzbekistan, Tashkent, Uzbekistan 3 Santa Fe Institute 399 Hyde Park Road, Santa Fe, NM 8750, USA 4 Centro Brasileiro de Pesuisas Fisicas Xavier Sigaud 50, Rio de Janeiro-RJ, Brazil Abstract The classic and the Lévy-Gnedenko central limit theorems play a key role in theory of probabilities, and also in Boltzmann-Gibbs (BG) statistical mechanics. They both concern the paradigmatic case of probabilistic independence of the random variables that are being summed. A generalization of the BG theory, usually referred to as nonextensive statistical mechanics and characterized by the index ( = recovers the BG theory), introduces global correlations between the random variables, and recovers independence for =. The classic central limit theorem was recently -generalized by some of us. In the present paper we - generalize the Lévy-Gnedenko central limit theorem. In Part I we described the -version of the -stable Lévy distributions. In Part II we study the (,, ) triplet, for which the mapping F : G G holds. This fact allows to study the corresponding attractors and to obtain a complete generalization of the -central limit theorem for random variables with infinite (2 )- variance. Introduction In the recent paper [], and in Part I [2] of the current paper, we discussed a -generalization of the classic central limit theorem applicable to the nonextensive statistical mechanics (see [3, 4, 5] and references therein), as well as a -generalization of the Lévy-Gnedenko central limit theorem. One of the important aspects of the obtained generalization is that it works for globally correlated random variables. In [] we introduced a new Fourier transform, i.e. the -Fourier transform and the function z(s) = + s 3 s to describe attractors of scaling limits of sums of -correlated random variables with a finite (2 )- variance 2. This description was essentially based on the mapping F : G G z(), () The -Fourier transform formally is defined as F [f](ξ) = f eixξ R dx and is a nonlinear operator if. 2 We reuired there < 2. Denoting Q = 2, it is easy to see that this condition is euivalent to the finitness of the Q-variance with Q < 3.

3 where G is the set of -Gaussians. In the current paper (consisting of two parts) we study a -analog of the -stable Lévy distributions. In this sense the present paper is a conceptual continuation of []. For simplicity we are dealing only with the standard symmetric one-dimensional case. Stable distributions with skewness and multivariate stable distributions can be studied in the same way applying the known classic techniues. Earlier studies devoted to the foundations of nonextensivity can be found in [5, 6, 7, 8, 9, 0,, 2], limit theorems in nonextensive statistical mechanics in [4, 3, 4, 5, 6], and various applications in [7, 8, 9, 20, 2, 22]. We study limits of sums of random variables with densities having the asymptotic behavior f C x + +( ), x, where 0 < < 2, 0 < < 2, and C > 0. These random variables have infinite Q-variance (Q = 2 ) for all 0 < < 2 if 0 < Q < 3 and for all < 2/( Q), if < Q < 0. The obtained limits represent a -generalization of the -stable Lévy distributions. In Part I we have given one of the possible classifications of these distributions in terms of their densities depending on the parameters Q and, where 0 < < 2. The classification was based on the mapping F : G L[2] G [], (2) where G [] is the set of all densities {be β ξ, b > 0, β > 0} and L = 3 + Q, Q = 2. + Note that the values of parameters Q and range in the set Q 0 = { < Q < 0, < 2 } {0 < Q < 3, 0 < < 2}, Q i.e. the case = 2, in the framework of this description, is peculiar. We note also that, for the -generalization of the Lévy-Gnedenko central limit theorem, the following lemma proved in Part I plays an essential role. Lemma. Let f(x), x R, be a symmetric probability density function of a given random vector. Further, let either (i) the (2 )-variance σ2 2 <, (associated with = 2), or (ii) f(x) C( x + +( ) ), with C > 0 and 0 < < 2, as x. Then for the -Fourier transform of f(x) the following asymptotic relation holds true: F [f](ξ) = µ, ξ + o( ξ ), ξ 0, (3) where µ, = 2 σ2 2 ν 2, if = 2; 2 2 (+( )) 2 0 Ψ (y) y + dy, if 0 < < 2, where ν 2 (f) = [f(x)]2 dx, and Ψ (y) = cos (2x). 3 (4) 3 For the definition of -cos see [, 8]. 2

4 In Part 2 we study a description of (, )-stable distributions differing from the description used in Part I. In the frame of the new description the value = 2 is no longer peculiar, but in this case we reuire. 4 More precisely, we expand the result of the paper [] to the region generalizing the mapping () in the form where Q = {(Q, ) : < Q < 3, 0 < 2}, F ζ() : G [] G z()[], 0 < < 2, 0 < 2, (5) ζ (s) = 2( ) and z (s) = + +. Note that if = 2, then ζ 2 () = and z 2 () = ( + )/(3 ), recovering the mapping (). 2 Description of the (, )-stable distributions based on the mapping (5) In this section we establish the general theorem, representing a family of assertions depending on the type of correlations. Note that the particular case, namely = 2 was proved in [], where the mapping F : G [2] G z() [2], /2 < < 2, was established. An importance of this mapping is its exactness in the sense that it transforms a -Gaussian into a z()-gaussian. For < 2, even in the classic case ( = ), there is no such exact mapping. Asymptotic representation of classic -stable Lévy distributions were obtained in [23, 24, 25, 26]. Our further arguments are also based on the asymptotic analysis. Let < < 2, or euivalently, < Q < 3, Q = 2. It follows from the definition of the - exponential that any density function g G [] has the asymptotic behavior g b x, b > 0, for large x. The set of all functions with this asymptotic we denote by B []. Obviously, G [] B []. At the same time, for any density f B [], there exists a uniue density g G [], such that f g, x. In this sense the two sets G [] and B [] are asymptotically euivalent (or asymptotically eual). Having this in mind we write (preferably) G [] instead of B []. Lemma 2. Let 0 < 2 be fixed. For arbitrary there exists 2 and a one-to-one mapping M, 2 such that M, 2 : G [] G 2 [2]. Obviously, if = 2, then = 2 and M, 2 is the identical operator. First we find the relationship between the three indices, and for which the mapping F : G [] (a) G [], (6) where (a) means that the mapping is in the sense of the asymptotic euivalence explained above, holds with (0, 2]. The exact meaning of (6) is M,z( ) F M, : G [] G []. In the case = 2, as we mentioned above, M, 2 = I, and the relationships = and = + 3 were found in [], giving (). 4 = Q = leads to the exponential functions, unlike to, which is connected asymptotically with power law functions. 3

5 Lemma 2.2 Assume 0 < 2 and that the numbers, and are connected with the relationships 2( ) = and = + (7) +. Then the mapping (6) holds true. Proof. Let f G (), which means that asymptotically f(x) b x /( ), x with some b > 0. Find the -Gaussian with the same asymptotics at. For G (β; x), (β > 0), we have the asymptotic euality G (β; x) x 2 Hence 2( ) = = Further, it follows from Corollary 2.0 of [], that where Taking into account the asymptotic euality we obtain, x. x F : G (2) G (2), 2( ). = + ( ) = 3 + ( ). G (β ; x) x = 2 + ( ) + x = Thus, the mapping (6) holds with and in Euation (7)., x, ( ) +. Let us introduce now two functions that are important for our further analysis: and z (s) = s + ( s) + s = ( s), 0 < 2, s < +, (8) + s 2( s) 2( s) ζ (s) = =, 0 < 2. (9) It can be easily verified that ζ (s) = s if = 2. The inverse, z (t), t (, ), of the the first function reads z t ( t) ( t) (t) = = ( t) + t. (0) The function z(s) possess the properties: z ( z ) = (s) s and z ( s ) =. If we denote z (s), = z () and, = z (), then z (, ) = and z ( ) =,. () 4

6 Corollary 2.3 The following mapping holds. F ζ() : G () (a) G z()(), < +, Corollary 2.4 There exists the following inverse -Fourier transform F ζ () : G z ()() (a) G (), < +, Using the above mentioned properties of the function z (s) we can derive a number of useful formulas for the -Fourier transforms. For instance, we get the mappings F ζ( F ζ( ) :, ) : G, () (a) G (), G () (a) G ()., The analogous formulas hold for the inverse -Fourier transforms as well. Introduce the seuence,n = z,n () = z(z,n ()), n =, 2,..., with a given = z 0 (), < +. We can extend the seuence,n for negative integers n =, 2,... as well putting, n = z, n () = z (z, n ()), n =, 2,.... It is not hard to verify that,n = ( ) + n( ) + n( ) =, n = 0, ±, ±2,... (2) + n( ) > + n for n < 0, and < + n for n > 0. (See Fig. ). Note that,n is a function only of (, n/), that,n for all n = 0, ±, ±2,..., if =, and that lim n ± z,n () = for all. E. (2) can be rewritten as follows: = + n, n = 0, ±, ±2,... (3),n This rewriting puts in evidence an interesting property. If we have a -Gaussian in the variable x /2 ( ), i.e., a -exponential in the variable x (whose asymptotic behavior is proportional to x ), its successive derivatives and integrations with regard to x precisely correspond to,n ),n -exponentials in the same variable x (whose asymptotic behavior is proportional to x 5. Along a similar line, it is also interesting to remark that E. (3) coincides with E. (3) of [27] (once we identify the present with the uantity z therein defined), therein obtained through a uite different approach (related to the renormalization of the index emerging from summing a specific expression over one degree of freedom). Let us note also that the definition of the seuence,n (E. (2)) can be given through the series of mappings 5 A typical illustration is as follows. Consider =, and a normalized -exponential distribution f(x) which identically vanishes for x < 0, and euals A(, β) e βx ( < 2, A(, β) > 0, β > 0) for x 0. The accumulated probability F ( x) = x f(x ) dx decreases from unity to zero when x increases from zero to infinity. This probability, freuently appearing in all kinds of applications, is given by F ( x) = A [ + ( (2 )β )βx] (2 )/( ), i.e., it is proportional to a,-exponential with, = +. 5

7 3 2, 2,0 2,2 2, 2,0 2 2,- 2,-2 2,-0 2,n 2, ,,0,2,,0,n 2,-,-2,-0, Figure : The -dependences of 2,n (top) and,n (bottom) as given by E. (2). We notice the tendency of all the n curves to collapse onto the,n = horizontal straight line if, and onto the = vertical straight line if =. This tendency is gradually intensified, n, when is fixed onto values decreasing from 2 towards zero. 6

8 Definition z, 2 z, z,0 = z, z,2 z... (4)... z z z, 2,,0 = z z z,,2... (5) Further, let us introduce the seuence,n = ζ(,n ). It is easy to see that or, euivalently,,n = 2( ) + n( ) 2,n + (n 2)( ) =, n = 0, ±,... (6) + n( ) = + n, n = 0, ±,... (7) It follows from Lemma 2.2 and definitions of seuences,n and,n that F,n : G,n () G,n+, n = 0, ±,... (8) Lemma 2.6 For all n = 0, ±, ±2,... there holds the following relations,n +,n+ = 2, (9) 2,n = 2,n. (20) Proof. Notice that,n+ = + 2( ) + (n )( ), which implies (9) immediately. The relation (20) can be checked easily. Remark 2.7 The property 2,n = 2,n shows that the seuences (2) and (6) coincide if = 2. Hence, the mapping (8) takes the form F 2,n : G 2,n (2) G 2,n+ (2), recovering Lemma 2.6 of []. Moreover, in this case the relation (9) holds for the seuence (2) as well. If < 2 6, then the values of the seuence (6) are splitted from the values of,n. The shift can be measured as,n,n = (2 )( ) + n( ), vanishing for = 2,, and for =,. Define for n = 0, ±,..., k =, 2,... the operators F k n (f) = F,n+k... F,n [f] = F,n+k [...F,n+ [F,n [f]]...], and Fn k (f) = F,n k... F,n [f] = F,n k [...F,n 2 [F,n [f]]...]. In addition, we assume that F k [f] = f, if k = 0 for any appropriate. Summarizing the above mentioned relationships, we obtain the following assertions. 6 As is known in the classic theory ( = ) this case describes anomalous diffusion processes. If =, then,n,n. In the nonextensive systems, as we can see from (8), there exist two separate seuences, which characterize the system under study. A physical confirmation of this theoretical result would be highly interesting. 7

9 Lemma 2.8 The following mappings hold:. F,n : G,n () (a) G,n+ (), n = 0, ±,...; 2. F k n : G,n () (a) G,k+n (), k =, 2,..., n = 0, ±, lim k ± F k n G () = G(), n = 0, ±,..., where G() is the set of classic -stable Lévy densities. Lemma 2.9 The series of mappings hold:... F, 2 G, () F, G () F F F,0,,2 G, () G,2 ()... (2)... F, 2 F F, F,0 F,,2 G, () G () G, () G,2 ()... (22) Theorem. Assume 0 < 2 and a seuence {...,, 2,,,,0,,,,2,...} as given in (4) with 0 = (/2, 2). Let X, X 2,..., X N,... be symmetric random variables mutually,k -correlated for some k and all having the same probability density function f(x) satisfying the conditions of Lemma.. (2 Then Z N, with D N (,k ) = (µ,k,n),k), is,k -convergent to a (,k, )-stable distribution, as N. Proof. The case = 2 coincides with Theorem of []. For k = 0, the first part of Theorem (-convergence) is proved in Part I of the paper. The same method is applicable for k. For reading convenience we reproduce the proof for arbitrary k = 0, ±,... Assume 0 < < 2. We evaluate F,k (Z N ). Denote Y j = D N () (X j ), j =, 2,... Then Z N = Y +...+Y N. It is not hard to verify that, for a given random variable X and real a > 0, there holds F [ax](ξ) = F [X](a 2 ξ ξ), for arbitrary. It follows from that F,k (Y ) = F,k [f]( ). (µ,k,n) Moreover, it follows from the,k -correlation of Y, Y 2,... (which is an obvious conseuence of the,k -correlation of X, X 2,...) and the associativity of the -product that ξ ξ F,k [Z N ](ξ) = F,k [f]( ) (µ,k,n),k...,k F,k [f]( ) (N factors). (23) (µ,k,n) Hence, making use of the properties of the -logarithm, from (23) we obtain ξ ln,k F,k [Z N ](ξ) = N ln,k F,k [f]( (µ,k,n) ) = N ln,k ( ξ N + o( ξ N )) = locally uniformly by ξ. Conseuently, locally uniformly by ξ, ξ + o(), N, (24) lim N F,k (Z N ) = e ξ,k G,k (). (25) Thus, Z N is,k -convergent. To show the second part of Theorem we use Lemma 2.9. In accordance with this lemma there exists a density f(x) G,k () that F,k [f] = e ξ,k. Hence, Z N is,k -convergent to a (,k, )-stable distribution, as N. Q.E.D. 8

10 3 Scaling rate analysis In paper [] we obtained the formula ( 3 k ) 2 β k = k 4 k C 2. (26) k 2 k for the Gaussian coefficient. It follows from this formula that the scaling rate in the case = 2 is δ = 2 k, (27) where k is the -index of the attractor. Moreover, if we insert the evolution parameter t, then the translation of a -Gaussian to a density in G () changes t to t 2/. Hence, applying these two facts to the general case, 0 < 2, and taking into account that the attractor index in our case is,k, we obtain the formula for the scaling rate δ = 2 (2,k (28) ). In accordance with Lemma 2.6 we have 2,k = /,k+. Conseuently, δ = 2,k+ = 2 + (k )( ) + (k + )( ). (29) Finally, in terms of Q = 2 the formula (29) takes the form δ = (k )( Q) 2 + (k + )( Q). (30) In the paper [] we noticed that the non-linear Fokker-Planck euation corresponds to the case k =. Taking this fact into account we can conjecture that the fractional generalization of the nonlinear Fokker-Planck euation is linked with the scaling rate δ = 2/(+ Q), which is derived from (30) putting k =. In the case = 2 we get the known result δ = 2/(3 Q) obtained in [4]. 4 About additive and multiplicative dualities In the nonextensive statistical mechanical literature, there are two transformations that appear uite freuently in various contexts. They are sometimes referred to as dualities. The multiplicative duality is defined through µ() = /, (3) and the additive duality is defined through ν() = 2. (32) They satisfy µ 2 = ν 2 =, where represents the identity, i.e., () =,. We also verify that (µν) m (νµ) m = (νµ) m (µν) m = (m = 0,, 2,...). (33) Consistently, we define (µν) m (νµ) m, and (νµ) m (µν) m. 9

11 Also, for m = 0, ±, ±2,..., and, (µν) m () = m (m ) m + m = + m( ) + m( ), (34) and ν(µν) m () = (µν) m µ() = m + 2 (m + ) m + m m + + m m + (m + ) = 2 + m( ) + m( ), (35) = m( ) m( ). (36) We can easily verify, from Es. (2) and (34), that the seuences 2,n (n = 0, ±2, ±4,...) and,n (n = 0, ±, ±2,...) coincide with the seuence (µν) m () (m = 0, ±, ±, 2,...). This establishes a remarkable connection between seuences which emerge naturally within the context of the - generalized central limit theorems, and the elementary dualities that we introduced above. However, its physical interpretation is yet to be found. It might be especially interesting if we take into account the fact that such a connection could be a crucial step (see footnote of page 5378 in [0]) for understanding the -triplet that was observed by NASA using data received from the spacecraft Voyager. Indeed, the existence of a -triplet, namely ( sen, rel, stat ), (related respectively to sensitivity to the initial conditions, relaxation, and stationary state) was conjectured in [2], and was indeed observed in the solar wind at the distant heliosphere [22]. 5 Conclusion The -CLT formulated in [] says that an appropriately scaled limit of sums of k -correlated random variables with a finite (2 k )-variance is a k -Gaussian, which is the k -Fourier image of a k - Gaussian. Here k and k are seuences defined as k = 2 + k( ), k = 0, ±,..., 2 + k( ) and k = k, k = 0, ±,... Schematically this theorem can be described as (see Fig. in []) {f : σ 2k (f) < } F k G k (2) F k G k (2) (37) where G (2) is the set of all -Gaussians. We have noted that the processes described by the -CLT can be effectively described by the triplet (P att, P cor, P scl ), where P att, P cor and P scl are parameters of attractor, correlation and scaling rate, respectively. We found that (P att, P cor, P scl ) ( k, k, k+ ). (38) In Part of this paper we have studied a -generalization of symmetric -stable Lévy distributions. Schematically the corresponding theorem (Theorem of [2]) is described as L(, ) F G () F G L(2), 0 < < 2, (39) 0

12 where L(, ) is the set of (, )-stable distributions, G L(2) is the set of L -Gaussians asymptotically euivalent to the densities f L(, ). The index L is linked with as follows L = L () = 3 + (2 ). + Note that the case = 2 is peculiar and we agree to refer to the scheme (37) in this case. In the present paper (Part 2, Theorem ) we have studied a -generalization of the CLT to the case when the (2 ) variance of random variables is infinite. The theorem that we have obtained generalizes the -CLT, which corresponds to = 2, to the full range 0 < 2. Schematically this theorem can be described as L(,k, ) F F,k,k G,k () G,k (2), 0 < 2, (40) having the same meaning as the scheme (37). The seuences,k and,k in this case read,k = + k( ), k = 0, ±,..., + k( ) and,k = 2( ), k = 0, ±,... + k( ) Note that the triplet (P att, P cor, P scl ) mentioned above takes, in this case, the form (P att, P cor, P scl ) (,k,,k, (2/),k+), which coincides with (38) if = 2. Finally, unifying the schemes (39) and (40) we obtain the general picture for the description of stable distributions: L(,k, ) F F,k,k G,k () G,k (2) (4) where F G L,k (2), L,k = L (,k ) = 3 + (2,k ) + In Fig. 2 the connection of (Q, ) Q with L and, (k = 0) is represented. If Q = and = 2 (the blue box in figure), then the random variables are independent in the usual sense and have finite variance. The standard CLT applies, and the attractors are classic Gaussians. If Q belongs to the interval (, 3) and = 2 (the blue straight line on the top), the random variables are not independent. If the random variables have a finite Q-variance, then the paper [] applies, and the attractors belong to the family of the -Gaussians. The support of attractors is compact if Q < and infinite if Q >. Note that runs in (, ) and in (, 5/3) if < Q < and < Q < 3, respectively. Thus, in this case, attractors ( -Gaussians) have finite classic variance (i.e., -variance) in addition to finite -variance. If Q = and 0 < < 2 (the vertical green line in the figure), we have the classic Lévy distributions), hence the random variables are independent, and have infinite variance. The classic.

13 Figure 2: (Q, )-regions (see the text). 2

14 Lévy-Gnedenko CLT applies, and the attractors belong to the family of the -stable Lévy distributions. If 0 < < 2, and Q belong to the interval (, 3) we observe the rich variety of situations that we have described. Let us first consider < Q < 3. Then the random variables are not independent, have infinite variance and infinite Q-variance. The rectangle { < Q < 3; 0 < < 2}, at the right of the classic Lévy line, is covered by non-intersecting curves C L {(Q, ) : 3 + Q + = L }, 5/3 < L < 3. Consistently with [], these families of curves describe all (Q, )-stable distributions based on the mapping (39) with -Fourier transform. The constant L is the index of the L -Gaussian attractor corresponding to the points (Q, ) on the curve C L. For example, the green curve corresponding to L = 2 describes all Q-Cauchy distributions, recovering the classic Cauchy-Poisson distribution if = (the green box in the figure). Every point (Q, ) laying on the brown curve corresponds to L = 2.5. At the same time, we have there another description of the (Q, )-stable distributions, which is based on the mapping (40) with -Fourier transform. Again the rectangle { < Q < 3; 0 < < 2} is covered by (straight) lines S {(Q, ) : 4 Q + 2 = 3 }, (42) which are obtained from (6) replacing n = and 2 = Q. For instance, every (Q, ) on the line F-I (the blue diagonal of the rectangle in the figure) identifies -Gaussians with = 5/3. This line is the frontier of points (Q, ) with finite and infinite classic variances. Namely, all (Q, ) above the line F-I identify attractors with finite variance, and points on this line and below identify attractors with infinite classic variance. Two bottom lines in Fig. 2 reflect the sets of corresponding to lines {( < Q < 3; = 2)} (the top boundary of the rectangle in the figure) and {( < Q < 3; = 0.6)} (the brown horizontal line in the figure). Finally, we describe the rectangle {(Q, ) : < Q < ; 0 < < 2)}, at the left of the classic Lévy line, analizing the formulas by continuity. In this region we see three frontier lines, F-II, F-III and F-IV. The line F-II splits the regions where the random variables have finite and infinite Q-variances. More precisely, the random variables corresponding to (Q, ) on and above the line F-II have a finite Q-variance, and, conseuently, the paper [] applies. Moreover, as seen in the figure, the L -attractors corresponding to the points on the line F-II are the classic Gaussians, because L = for these (Q, ). It follows from this fact, that L -Gaussians corresponding to points above F-II have compact support (the blue region in the figure), and L -Gaussians coresponding to points on this line and below have infinite support. The line F-III splits the points (Q, ) whose L -attractors have finite or infinite classic variances. More precisely, the points (Q, ) above this line identify attractors (in terms of L -Gaussians) with finite classic variance, and the points on this line and below identify atttractors with infinite clasic variance. The frontier line F-IV with the euation Q + 2 = 0 and joining the points (, 0) and (, ) is related to attractors in terms of -Gaussians. It follows from (42) that for (Q, ) laying on the line F-IV, the index =. Thus the horizontal lines corresponding to < can be continued only up to the line F-IV with (, 3 4 Q+2 ) (see the dashed horizontal brown line in the figure). If 0, the Q-interval becomes narrower, but -interval becomes larger tending to (, 3). Let us stress that Fig. 2 corresponds to the case k = 0 in the description (4). The cases k 0 can be analyzed in the same way. We also note that the method we used for the -generalization of 3

15 the classic and the Lévy-Gnedenko CLTs is applicable for 0 < < 2 (or euivalently < Q < 3). So, the description of (Q, )-stable distributions corresponding to the region Q (the white rectangle and its left side in the figure) remains open at this point. Acknowledgments We acknowledge thoughtful remarks by R. Hersh and E.P. Borges. Financial support by the Fullbright Foundation, SI International and AFRL (USA) are acknowledged as well. References [] S. Umarov, C. Tsallis and S.Steinberg, A generalization of the central limit theorem consistent with nonextensive statistical mechanics ArXiv(2006) [2] S. Umarov, C. Tsallis, M. Gell-Mann and S.Steinberg, A -generalization of the Lévy- Gnedenko central limit theorem. Part I: -analogs of symmetric -stable distributions [3] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (988). See also E.M.F. Curado and C. Tsallis, Generalized statistical mechanics: connection with thermodynamics, J. Phys. A 24, L69 (99) [Corrigenda: 24, 387 (99) and 25, 09 (992)], and C. Tsallis, R.S. Mendes and A.R. Plastino, The role of constraints within generalized nonextensive statistics, Physica A 26, 534 (998). [4] C. Tsallis, Nonextensive statistical mechanics, anomalous diffusion and central limit theorems, Milan Journal of Mathematics 73, 45 (2005). [5] M. Gell-Mann and C. Tsallis, Nonextensive Entropy - Interdisciplinary Applications (Oxford University Press, New York, 2004). [6] D. Prato and C. Tsallis, Nonextensive foundation of Levy distributions, Phys. Rev. E 60, 2398 (999), and references therein. [7] L. Nivanen, A. Le Mehaute and Q.A. Wang, Generalized algebra within a nonextensive statistics, Rep. Math. Phys. 52, 437 (2003). [8] E.P. Borges, A -generalization of circular and hyperbolic functions, Physica A: Math. Gen. 3, 528 (998). [9] E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340, 95 (2004). [0] C. Tsallis, M. Gell-Mann and Y. Sato, Asymptotically scale-invariant occupancy of phase space makes the entropy S extensive, Proc. Natl. Acad. Sc. USA 02, 5377 (2005). [] J. Marsh and S. Earl, New solutions to scale-invariant phase-space occupancy for the generalized entropy S, Phys. Lett. A 349, 46 (2005). [2] C. Tsallis, M. Gell-Mann and Y. Sato, Extensivity and entropy production, Europhysics News 36, 86 (2005). 4

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