Central limit theorem and stable laws for intermittent maps

By S. Gouëzel. 2002

122 citations recorded by the source corpus.

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+ α , for α ∈ (0 , 1). In particular, for α > 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) 6 = 0. The proof uses spectral techniques introduced by Sarig, and Wiener’s Lemma in non-commutative Banach algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa5ce5491b5a4dee55a43d66e7b864f394075906

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Homogenization for deterministic maps and multiplicative noise (2013, background): “(More precisely, it follows from Gouëzel [28] that if c ∈ Rd and c · f0(0) = 0, then n−γ ∑n−1 j=0 c · f0(y(j)) converges in distribution to a one-dimensional”
On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks (2008, background): “This can be compared to the intermittency property of the dynamics of certain maps of the interval, with a neutral fixed point (see, for example, [11]).”
Statistical limit theorems for suspension flows (2004, result): “This provides a significantly simplified proof of results in Gouëzel [17].”
Stable limits for sums of dependent infinite variance random variables (2009, methodology): “We would like to thank the referee for bringing the paper by Gouëzel (29) to our attention.”
Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains (2009, methodology): “This spectral control makes it possible to apply Theorem 1.1 in Gouëzel (2004b), dealing with renewal sequences of operators as above.”
Spectral Gap and Transience for Ruelle Operators on Countable Markov Shifts (2009, methodology): “Next we study the dierence ht h, as in [ G1 ].”