Superposition of chaotic process with convergence to Levy's stable law

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