Statistical-Mechanical Foundation of the Ubiquity of Lévy Distributions in Nature

We show that the use of the recently proposed thermostatistics based on the generalized entropic form $S_q\equiv \frac{k(1-\Sigma i^{}{p}_i^q)}{\mathrm{}(q-1\mathrm{})\mathrm{}}$ (where $q\in R$, with $q=1\mathrm{}$ corresponding to the Boltzmann-Gibbs-Shannon entropy $-k\Sigma i^{}{p}_i \ln p_i$), together with the Lévy-Gnedenko generalization of the central limit theorem, provide a basic step towards the understanding of why Lévy distributions are ubiquitous in nature. A consistent experimental verification is proposed.