Rates in almost sure invariance principle for slowly mixing dynamical systems
2019 ◽
Vol 40
(9)
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pp. 2317-2348
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Keyword(s):
The One
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We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.
1991 ◽
Vol 3
(3)
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pp. 361-379
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Keyword(s):
2019 ◽
Vol 575
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pp. 1145-1154
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Keyword(s):
2008 ◽
Vol 18
(12)
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pp. 3625-3632
1994 ◽
Vol 26
(01)
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pp. 80-103
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