In this report, new almost sure convergence properties for vectorial martingale transforms are established. Assuming some regularity conditions both on the increasing process and on the moments of the martingale, one can show that normalized moments of any even order do also converge in the almost sure cental limit theorem for martingales. Firm conjectures are proposed to relax some of the hypotheses, the goal being to cover a much wider class of processes. Cases studies are proposed, including linear models and some branching processes, where new asymptotic estimation and prediction errors are obtained.