Summary
Let T be a measure preserving transformation on a finite measure space. Then for certain increasing sequences k 1 k 2, ... of positive integers, called uniform sequences, the average
converges in the mean and almost everywhere. For strongly mixing transformations and any sequence of powers, an individual ergodic theorem with weights is valid.
Article PDF
Similar content being viewed by others
Literature
Blum, J.R., and D.L. Hanson: On the mean ergodic theorem for subsequences. Bull. Amer. math. Soc. 66, 308–311 (1960).
Fomin, S.: On dynamical systems with pure point spectrum. Doklady Akad. Nauk SSSR 77, 29–32 (1951).
Halmos, P.R.: Lectures on ergodic theory. Math. Soc. Japan (1953).
Jacobs, K.: Neuere Methoden und Ergebnisse der Ergoden theorie. Springer Erg. d. Math. (1960).
Krengel, U.: Classification of states for operators. Proc. V Berkeley Sympos math. Statist. Probab. II, 2 (1967).
Oxtoby, J.C.: Ergodic sets. Bull. Amer. math. Soc. 58, 116–136 (1952).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brunel, A., Keane, M. Ergodic theorems for operator sequences. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 231–240 (1969). https://doi.org/10.1007/BF00534842
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00534842