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.
"Law of the iterated logarithm for stationary processes." Ann. Probab. 36 (1) 127 - 142, January 2008. https://doi.org/10.1214/009117907000000079