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.


60F15 Strong limit theorems
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