Abstract
A necessary and sufficient condition for the almost everywhere convergence of the “moving” ergodic averages\((\Phi (n))^{ - 1} \mathop \Sigma \limits_{i = n - \Phi (n) + 1}^n x_E (T^i x)\) is given. The result is then generalized to ergodic flows, and finally constrasted with earlier results ofPfaffelhuber andJain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akcoglu, M.A., andA. del Junco: Convergence of Averages of Point Transformations, Proc. Amer. Math. Soc.49 (1), 1975, 265–266.
Ambrose, W., andS. Kakutani: Structure and Continuity of measurable flows, Duke Math. J.9, 1942, 25–42.
Bahadur, R.R., andR. Ranga Rao: On deviations of the sample mean, Annals of Mathematical Statistics31, 1960, 1015–1027.
Belley, J.M.: Invertible measure preserving transformations are pointwise convergent, Proc. Amer. Math. Soc.43, 1974, 159–162.
Cramér, H.: Sur un Nouveau théorèm-limite de la theorie des probabilités. Actualités Scientifiques et Industrielles, No 736, Herman et Cie, Paris, 1938.
Jain, N.C.: Tail Probabilities for Sums of Independent Random Variables, Z. für Wahrscheinlichkeitstheorie verw. Gebiete33, 1975, 155–166.
Halmos, P.R.: Lectures on Ergodic Theory, Chelsea Publishing Co., New York 1956.
Pfaffelhuber, E.: Moving Shift Averages for Ergodic Transformations, Metrika22, 1975, pp. 97–101.
Rudolf, D.: Some Problems of Ergodic Flows, Stanford University Thesis, Stanford C.A. 1975.
Smorodinky, M.: Ergodic theory, entropy, Lecture Notes in Mathematics, Vol. 214, New York 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
del Junco, A., Steele, J.M. Moving averages of ergodic processes. Metrika 24, 35–43 (1977). https://doi.org/10.1007/BF01893390
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01893390