We analyze here the asymptotic convergence properties of a class of splitting algorithms for finding a zero of the sum of two maximal monotone operators, not necessarily differentiable. We use the concept of proto-differ-entiability to obtain some new bounds on the global error of the sequence of iterates and explain the bad spiraling effect already observed in practice. After linking our model with the Lawrence-Spingarn's folding operators, we show how to accelerate convergence by skipping the averaging steps in a particular way. Numerical results on medium-term stochastic planning problems confirm the nice behavior of the modified algorithm.