The central limit theorem for monotone convolution with applications to free Levy processes and infinite ergodic theory Jiun-Chau Wang 46L5446L5328D0560F05Monotone convolutionfree convolutioncentral limit theoremfree Levy processinfinite ergodic theoryinner function Using free harmonic analysis and the theory of regular variation, we show that the monotonic strict domain of attraction for the standard arc-sine law coincides with the classical one for the standard normal law. This leads to the most general form of the monotonic central limit theorem and a complete description for the asymptotics of the norming constants. These results imply that the L\'evy measure for a centered free L\'evy process of the second kind cannot have a slowly varying truncated variance. In particular, the second kind of free L\'evy processes with zero means and finite variances do not exist. Finally, the method of proofs allows us to construct a new class of conservative ergodic measure preserving transformations on the real line $\mathbb{R}$ equipped with Lebesgue measure, showing an unexpected connection between free analysis and infinite ergodic theory for inner functions. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5229 10.1512/iumj.2014.63.5229 en Indiana Univ. Math. J. 63 (2014) 303 - 327 state-of-the-art mathematics http://iumj.org/access/