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

Pointwise ergodic theorems with rate and application to limit theorems for stationary processes