1. For certain weakly dependent processes which permit a spectral gap for the characteristic function operator, local limit theorems are derived when the one-dimensional marginals belong to the domain of attraction of a stable law. These processes occur in the study of Poincare series of the Z-extension of the three punctured sphere, and the local limit theorem permits to calculate the Poincare exponent.
2. The classical de Moivre Local Limit Theorem is extended to an almost sure version (along the lines of almost sure central limit theorems).
