## Invariance principles under the Maxwell-Woodroofe condition in Banach spaces.(English)Zbl 1374.60060

The author studies stationary processes with values in various Banach spaces and specially $$L^p$$-spaces with $$1\leq p< \infty$$. He shows that the condition given by M. Maxwell and M. Woodroofe [Ann. Probab. 28, No. 2, 713–724 (2000; Zbl 1044.60014)] is sufficient for the law of the iterated logarithm and optimal in some sense. The almost sure invariance principle and the weak invariance principle are obtained for large classes of Banach spaces. The tools in the proofs are maximal inequalities and martingale approximation. Several examples are given including the case of empirical processes.

### MSC:

 60F17 Functional limit theorems; invariance principles 60F25 $$L^p$$-limit theorems 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 37A50 Dynamical systems and their relations with probability theory and stochastic processes

Zbl 1044.60014
Full Text: