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Law of the iterated logarithm for stationary processes

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Ou Zhao, Michael Woodroofe

Probability · Submitted 24 Dec 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv IDmath/0612747 (version 2)
Pith ID3S6N3I4T, Pith's permanent record of this paper and its review
CitationMichael Woodroofe, Ou Zhao. "Law of the iterated logarithm for stationary processes." arXiv:math/0612747 (2006). https://pith.science/paper/3S6N3I4T
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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.69 · compares with prior results 0.44 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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 $E(R_n^2)=o(n)$. background 1.00 0.05 0.03 asserted 0.96
s1 Here we explore the law of the iterated logarithm (LIL) for the same class of processes. contribution 0.99 0.83 0.07 asserted 0.55
s2 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. result 0.80 0.70 0.07 asserted 1.00
s3 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. result 0.93 0.88 0.59 asserted 1.00
s4 As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. result 0.88 0.94 0.13 asserted 1.00
s5 Invariance principles are obtained as well. result 0.86 0.90 0.06 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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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