Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product
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چکیده
In this article we review the standard versions of the Central and of the Lévy-Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution Gq (X) ≡ Aq [ 1 +(q−1)Bq (X − μ̄q) ] 1 1−q (Aq > 0; Bq > 0; q < 3), known as q-Gaussian. This distribution emerges upon extremisation of the nonadditive entropy Sq ≡ k (1− ∫ [p(X)] dX)/(1−q), basis of nonextensive statistical mechanics. It has a finite variance for q < 5/3, and an infinite one for q ≥ 5/3. We exhibit that, in the case of (standard) independence, the q-Gaussian has either the Gaussian (if q < 5 3 ) or the α-stable Lévy distributions (if q > 5 3 ) as its attractor in probability space. Moreover, we review a generalisation of the product, the q-product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory.
منابع مشابه
Convergence of the probability of large deviations in a model of correlated random variables having compact-support Q-Gaussians as limiting distributions
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