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−3/2‖E(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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