Should I stay or should I go? Zero-size jumps in random walks for Levy flights

We study Markovian continuous-time random walk models for Levy 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 Levy-like motion for wild animals.