  We study Markovian continuous-time random walk models for L\'evy flights and
we show an example in which the convergence to stable densities is not
guaranteed when jumps follow a bi-modal power-law distribution that is equal to
zero in zero. The significance of this result is two-fold: i) with regard to
the probabilistic derivation of the fractional diffusion equation and also ii)
with regard to the concept of site fidelity in the framework of L\'evy-like
motion for wild animals.
