Classification of random trajectories based on the fractional Lévy stable motion

•We propose a new approach for the analysis of experimental trajectories based on the fractional Lévy stable motion.•It takes into account non-Gaussian, stable distributions and the memory parameter, which governs the diffusion type.•We apply the procedure to the analysis of the Golding-Cox mRNA dataset (I. Golding et al., Cell 123, 1025, 2005).•The methods are universal and can be applied to any experimental data which exhibit a non-Gaussian behavior.