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On the central limit theorem for dynamical systems


Authors: Robert Burton and Manfred Denker
Journal: Trans. Amer. Math. Soc. 302 (1987), 715-726
MSC: Primary 60F05; Secondary 28D05
DOI: https://doi.org/10.1090/S0002-9947-1987-0891642-6
MathSciNet review: 891642
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Abstract: Given an aperiodic dynamical system $ (X,T,\mu )$ then there is an $ f \in {L^2}(\mu )$ with $ \smallint fd\mu = 0$ satisfying the Central Limit Theorem, i.e. if $ {S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}$ and $ {\sigma _m} = {\left\Vert {{S_m}f} \right\Vert _2}$ then

$\displaystyle \mu \left\{ {x\vert\frac{{{S_m}f(x)}}{{{\sigma _m}}} < u} \right\... ...fty }^u {{\text{exp}}} \left[ {\frac{{ - {\upsilon ^2}}}{2}} \right]d\upsilon .$

The analogous result also holds for flows.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0891642-6
Keywords: Central limit theorem, dynamical systems
Article copyright: © Copyright 1987 American Mathematical Society

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