We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.
Several interpolation theorems on martingale Hardy spaces over weighted measure spaces are given. Our proofs are based on
the atomic decomposition of martingale Hardy spaces over weighted measure spaces. As applications of interpolation theorems,
some inequalities of martingale transform operator are obtained.