Law of the iterated logarithm for stationary processes

Details Law of the iterated logarithm for stationary processes Law of the iterated logarithm for stationary processes

2006-12-24

arxiv.org/abs/math/0612747v2

Annals of Probability 2008, Vol. 36, No. 1, 127-142

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000079 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000079

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.

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