## The Annals of Probability

### On the Individual Ergodic Theorem for $K$-Automorphisms

#### Abstract

Let $(X, \mathscr{B}(X), P)$ be a probability space and let $T$ be a $K$-automorphism. If $T$ satisfies a Rosenblatt mixing condition of a certain kind, we show that if $\{k_n\}^\infty_{n=1}$ is an arbitrary increasing sequence of integers and $g$ belongs to a certain class of functions then $$\lim_{n\rightarrow\infty} \frac{1}{n} \sum^n_{j=1} g(T^{k_j}x) = E(g) \mathrm{a.s.}$$

#### Article information

Source
Ann. Probab., Volume 5, Number 2 (1977), 309-314.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995857

Digital Object Identifier
doi:10.1214/aop/1176995857

Mathematical Reviews number (MathSciNet)
MR430207

Zentralblatt MATH identifier
0368.28025

JSTOR
Blum, J. R.; Reich, J. I. On the Individual Ergodic Theorem for $K$-Automorphisms. Ann. Probab. 5 (1977), no. 2, 309--314. doi:10.1214/aop/1176995857. https://projecteuclid.org/euclid.aop/1176995857