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Transactions of the American Mathematical Society

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A pointwise ergodic theorem for the group of rational rotations


Authors: Lester E. Dubins and Jim Pitman
Journal: Trans. Amer. Math. Soc. 251 (1979), 299-308
MSC: Primary 60G42; Secondary 28D99
DOI: https://doi.org/10.1090/S0002-9947-1979-0531981-7
MathSciNet review: 531981
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Abstract: Let f be a bounded, measurable function defined on the multiplicative group $ \Omega $ of complex numbers of absolute value 1, and define

$\displaystyle {{f_n}(\omega ) = \frac{1} {n}\sum\limits_{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,$ ($ (1)$)

where $ {z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $ n(k)$ is an increasing sequence of positive integers with $ n(k)$ dividing $ n(k')$ whenever $ k < k'$, then $ {f_{n(k)}}$ converges almost surely as $ k \to \infty $.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0531981-7
Keywords: Martingales, ergodic theory, permutable groups, conditional independence
Article copyright: © Copyright 1979 American Mathematical Society

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