Rates of convergence of non-extensive statistical distributions to Lévy distributions in full and half-spaces

The Lévy-type distributions are derived using the principle of maximum Tsallis non-extensive entropy both in the full and half-spaces. The rates of convergence to the exact Lévy stable distributions are determined by taking the N -fold convolutions of these distributions. The marked difference between the problems in the full and half-spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Lévy indices. An important result emerging from the present analysis is deduced if interpreted in terms of random walks, implying the dependence of the asymptotic long-time behaviours of the walks on the ranges of the Lévy indices if N is identified with the total time of the walks.