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December, 1977 On the Individual Ergodic Theorem for Subsequences
Jakob I. Reich
Ann. Probab. 5(6): 1039-1046 (December, 1977). DOI: 10.1214/aop/1176995673


We define a concept of saturation for a sequence of integers $\{k_j\}$. In the main theorem we prove that if $\{k_j\}$ saturates and $T$ is any weakly mixing measure-preserving transformation on an arbitrary probability space, then there exists a dense set $\mathscr{D}_T \subset L^2$ such that for $f \in \mathscr{D}_T$ $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j}x) = E(f) \mathrm{a.e.}$$ This has the following application to probability theory: Let $Y_1, Y_2,\cdots$ be independent and identically distributed positive (or negative) integer-valued random variables with $E(Y_1) < \infty$. Let $$k_j(\omega) = \sum^j_{l=1} Y_l(\omega) \quad j = 1,2,\cdots$$. Then there exists a set $C$ of probability one such that for $\omega \in C$ and for any weakly mixing measure preserving transformation $T$ on an arbitrary probability space $$\lim_{N\rightarrow\infty} \frac{1}{N} \sum^N_{j=1} f(T^{k_j(\omega)}x) = E(f) \mathrm{a.e.}$$ for all $f \in L^1$.


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Jakob I. Reich. "On the Individual Ergodic Theorem for Subsequences." Ann. Probab. 5 (6) 1039 - 1046, December, 1977.


Published: December, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0372.60054
MathSciNet: MR444906
Digital Object Identifier: 10.1214/aop/1176995673

Primary: 60G17
Secondary: 60B10 , 60G50

Keywords: subsequences , Sums of independent random variables , The individual ergodic theorem

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 6 • December, 1977
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