  There has been recent interest in the conditional central limit question for
(strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial
sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square
integrable martingale with stationary increments and $R_n$ is a remainder term
for which $E(R_n^2)=o(n)$. Here we explore the law of the iterated logarithm
(LIL) for the same class of processes. Letting $\Vert\cdot\Vert$ denote the
norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary
process to have the form $S_n=M_n+R_n$ is that $n^{-3/2}\Vert
E(S_n|X_0,X_{-1},...)\Vert$ be summable. A sufficient condition for the LIL is
only slightly stronger, requiring $n^{-3/2}\log^{3/2}(n)\Vert
E(S_n|X_0,X_{-1},...)\Vert$ 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.
