  We consider a previously devised model describing Levy random walks (Phys.
Rev E 79, 011110; 80, 031148, (2009)). It is demonstrated numerically that the
given model describes Levy random walks with superdiffusive, ballistic, as well
as superballistic dynamics. Previously only the superdiffusive regime has been
analyzed. In this model the walker velocity is governed by a nonlinear Langevin
equation. Analyzing the crossover from small to large time scales we find the
time scales on which the velocity correlations decay and the walker motion
essentially exhibits Levy statistics. Our analysis is based on the analysis of
the geometric means of walker displacements and allows us to tackle probability
density functions with power-law tails and, correspondingly, divergent moments.
