  We obtain pointwise ergodic theorems with rate under conditions expressed in
terms of the convergence of series involving $\|\sum_{k=1} ^nf\circ
\theta^k\|_2$, improving previous results. Then, using known results on
martingale approximation, we obtain some LIL for stationary ergodic processes
and quenched central limit theorems for functional of Markov chains. The proofs
are based on the use of the spectral theorem and, on a recent work of
Zhao-Woodroofe extending a method of Derriennic-Lin.
