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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the central limit theorem for dynamical systems
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by Robert Burton and Manfred Denker PDF
Trans. Amer. Math. Soc. 302 (1987), 715-726 Request permission

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 ${\sigma _m} = {\left \| {{S_m}f} \right \|_2}$ then \[ \mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} < u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .\] The analogous result also holds for flows.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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