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q-generalization of symmetric alpha-stable distributions. Part II | | Sabir Umarov
; Constantino Tsallis
; Murray Gell-Mann
; Stanly Steinberg
; | | Date: |
1 Jun 2006 | | Subject: | Statistical Mechanics | | Abstract: | The classic and the L’evy-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 $q$ ($q=1$ recovers the BG theory), introduces global correlations between the random variables, and recovers independence for $q=1$. The classic central limit theorem was recently $q$-generalized by some of us. In the present paper we $q$-generalize the L’evy-Gnedenko central limit theorem. In Part I we described the $q$-version of the $alpha$-stable L’evy distributions. In Part II we study the $(q^{ast},q,q_{ast})-$triplet, for which the mapping $F_{q^{ast}}: , mathcal{G}_{q}
ightarrow mathcal{G}_{q_{ast}}$ holds. This fact allows to study the corresponding attractors and to obtain a complete generalization of the $q$-central limit theorem for random variables with infinite $(2q-1)$-variance. | | Source: | arXiv, cond-mat/0606040 | | Services: | Forum | Review | PDF | Favorites | |
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