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January 2008 Law of the iterated logarithm for stationary processes
Ou Zhao, Michael Woodroofe
Ann. Probab. 36(1): 127-142 (January 2008). DOI: 10.1214/009117907000000079


There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes …, X−1, X0, X1, … whose partial sums Sn=X1+⋯+Xn are of the form Sn=Mn+Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(Rn2)=o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting ‖⋅‖ denote the norm in L2(P), a sufficient condition for the partial sums of a stationary process to have the form Sn=Mn+Rn is that n−3/2E(Sn|X0, X−1, …)‖ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n−3/2log3/2(n)‖E(Sn|X0, X−1, …)‖ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.


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Ou Zhao. Michael Woodroofe. "Law of the iterated logarithm for stationary processes." Ann. Probab. 36 (1) 127 - 142, January 2008.


Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1130.60039
MathSciNet: MR2370600
Digital Object Identifier: 10.1214/009117907000000079

Primary: 60F15
Secondary: 60F05

Keywords: Conditional central limit question , ergodic theorem , Fourier series , Markov chains , Martingales , operators on L^2

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 1 • January 2008
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