Open Access
May 2017 Invariance principles under the Maxwell–Woodroofe condition in Banach spaces
Christophe Cuny
Ann. Probab. 45(3): 1578-1611 (May 2017). DOI: 10.1214/16-AOP1095


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.


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Christophe Cuny. "Invariance principles under the Maxwell–Woodroofe condition in Banach spaces." Ann. Probab. 45 (3) 1578 - 1611, May 2017.


Received: 1 March 2015; Revised: 1 January 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1374.60060
MathSciNet: MR3650410
Digital Object Identifier: 10.1214/16-AOP1095

Primary: 60B12 , 60F17 , 60F25
Secondary: 37A50

Keywords: Banach valued processes , compact law of the iterated logarithm , Invariance principles , Maxwell–Woodroofe’s condition

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • May 2017
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