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 Sₐ (1-{i^p₈^q) } (q-1) (where q, with q=1 corresponding to the Boltzmann-Gibbs-Shannon entropy -ki^p₈ ln p₈), together with the L\'evy-Gnedenko generalization of the central limit theorem, provide a basic step towards the understanding of why L\'evy distributions are ubiquitous in nature. A consistent experimental verification is proposed.