## Law of the iterated logarithm for stationary processes.(English)Zbl 1130.60039

Summary: There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $$\dots, X_{-1}, X_0, X_1,\dots$$ whose partial sums $$S_n=X_1+\dots +X_n$$ 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 $$\|\cdot\|$$ 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 $$n^{-3/2}\|E(S_n|X_0, X_{-1},\dots)\|$$ be summable. A sufficient condition for the LIL is only slightly stronger, requiring $$n^{-3/2}\log^{3/2}(n)\|E(S_n|X_0, X_{-1},\dots)\|$$ 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.

### MSC:

 60F15 Strong limit theorems
Full Text:

### References:

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