In this thesis we study the speed of convergence in the Boolean and monotone central limit theorems. We also investigate some ergodic properties for certain transformations on the real line. In the Boolean central limit theorem, we obtain a bound of order n^(-1/3) of the speed of convergence for measures with finite fourth moment. The proof is based on the Boolean cumulants. When the measures have bounded support, we get a bound of order n^(-1/2) of the speed of convergence, and we show that this bound is sharp. We derive this result from a more general theorem describing very explicitly the convergence in the Boolean central limit theorem. These results are in terms of the Lévy metric. For the monotone central limit theorem, we obtain a bound of order n^(-1/8) of the speed of convergence for measures with finite fourth moment. We improve this bound for the case of measures with finite sixth moment. We get a bound of order n^(-1/4), and we prove that this bound is sharp. The proofs of these results are based on the complex analysis methods of non-commutative probability. Finally, the F-transform of measures singular to the Lebesgue measure induce a transformation on the real line which preserves the Lebesgue measure. We prove that for measures of zero mean and unit variance these transformations are pointwise dual ergodic with return sequence pi(2n)^(1/2), extending a previous result of Aaronson.