Invariance principles under the Maxwell.Woodroofe condition in Banach spaces

We prove that, for (adapted) stationary processes, the so-called Maxwell.Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of Lp-valued random variables, with 1.p<.. In this setting, we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [Ann. Probab. 33 (2005) 798.815]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.