Superposition of chaotic processes with convergence to Lévy’s stable law

We construct a family of chaotic dynamical systems with explicit distributions with broad tails, which always violate the central limit theorem. In particular, we show that the superposition of many statistically independent, identically distributed random variables obeying such a chaotic process converge in density to L\'evy's stable laws in a full range of index parameters in a unified manner. The theory related to the connection between deterministic chaos and non-Gaussian distributions gives us a systematic view of the purely mechanical generation of L\'evy's stable laws.