## The Annals of Probability

### Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

Christophe Cuny

#### Abstract

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 $L^{p}$-valued random variables, with $1\le p<\infty$. 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.

#### Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1578-1611.

Dates
Revised: January 2016
First available in Project Euclid: 15 May 2017

https://projecteuclid.org/euclid.aop/1494835226

Digital Object Identifier
doi:10.1214/16-AOP1095

Mathematical Reviews number (MathSciNet)
MR3650410

Zentralblatt MATH identifier
1374.60060

#### Citation

Cuny, Christophe. Invariance principles under the Maxwell–Woodroofe condition in Banach spaces. Ann. Probab. 45 (2017), no. 3, 1578--1611. doi:10.1214/16-AOP1095. https://projecteuclid.org/euclid.aop/1494835226

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