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Central limit theorem and stable laws for intermittent maps

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Sebastien Gouezel

Dynamical Systems · Submitted 6 Nov 2002

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

The authors' main claim In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws.

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arXiv IDmath/0211117 (version 2)
Pith IDSRQMARI4, Pith's permanent record of this paper and its review
CitationSebastien Gouezel. "Central limit theorem and stable laws for intermittent maps." arXiv:math/0211117 (2002). https://pith.science/paper/SRQMARI4
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,681 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.52 · compares with prior results 0.07 · mentions code or data availability 0.01.

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

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+\alpha}$, for $\alpha\in (0,1)$. In particular, for $\alpha>1/2$, we show that the Birkhoff sums of a H\"older observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. contribution 0.80 0.94 0.81 asserted 1.00
s1 They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+\alpha}$, for $\alpha\in (0,1)$. contribution 0.50 0.38 0.07 asserted 0.99
s2 In particular, for $\alpha>1/2$, we show that the Birkhoff sums of a H\"older observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. result 0.99 0.95 0.56 asserted 1.00
s3 The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras. method 1.00 0.51 0.03 asserted 0.99

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The paper
Original abstract

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form $x+x^{1+\alpha}$, for $\alpha\in (0,1)$. In particular, for $\alpha>1/2$, we show that the Birkhoff sums of a H\"older observable $f$ converge to a normal law or a stable law, depending on whether $f(0)=0$ or $f(0)\not=0$. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in noncommutative Banach algebras.

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