Limit theorems under the Maxwell-Woodroofe condition in Banach spaces

Understand

We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense.

• We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers.
• Those results actually hold in the context of Banach valued stationary processes, including the case of $L^r$-valued random variables, with $1\le r<\infty$.
• In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \cite{PU}.