The Annals of Probability

On the Individual Ergodic Theorem for Subsequences

Jakob I. Reich

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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 $f \in L^1$.

Article information

Ann. Probab., Volume 5, Number 6 (1977), 1039-1046.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G17: Sample path properties
Secondary: 60G50: Sums of independent random variables; random walks 60B10: Convergence of probability measures

The individual ergodic theorem subsequences sums of independent random variables


Reich, Jakob I. On the Individual Ergodic Theorem for Subsequences. Ann. Probab. 5 (1977), no. 6, 1039--1046. doi:10.1214/aop/1176995673.

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