The Annals of Probability

Law of the iterated logarithm for stationary processes

Ou Zhao and Michael Woodroofe

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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.

Article information

Ann. Probab., Volume 36, Number 1 (2008), 127-142.

First available in Project Euclid: 28 November 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

Conditional central limit question ergodic theorem Fourier series martingales Markov chains operators on L^2


Zhao, Ou; Woodroofe, Michael. Law of the iterated logarithm for stationary processes. Ann. Probab. 36 (2008), no. 1, 127--142. doi:10.1214/009117907000000079.

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