×

Polynomially moving ergodic averages. (English) Zbl 0642.28008

Summary: Given an increasing sequence of positive integers \(\{m_ n\}\), a non- decreasing sequence of positive integers \(\{b_ n\}\), and a measurable, measure-preserving ergodic transformation \(\tau\) on a probability space (\(\Omega\),\({\mathcal F},\mu)\), the a.s. convergence of the moving averages \(T_ n(f)=b_ n^{-1}\sum^{m_ n+b_ n}_{k=m_ n+1}f(\tau \quad k)\) is considered, for \(f\in L_ p(\Omega)\). A counterexample is constructed in the case of polynomial-like \(\{m_ n\}\).

MSC:

28D05 Measure-preserving transformations
60F15 Strong limit theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. A. Akcoglu and A. del Junco, Convergence of averages of point transformations, Proc. Amer. Math. Soc. 49 (1975), 265 – 266. · Zbl 0278.28011
[2] Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), no. 3, 185 – 197. · Zbl 0398.28021 · doi:10.1007/BF01673506
[3] A. del Junco and J. M. Steele, Moving averages of ergodic processes, Metrika 24 (1977), no. 1, 35 – 43. · Zbl 0357.28017 · doi:10.1007/BF01893390
[4] Joseph Rosenblatt, Ergodic group actions, Arch. Math. (Basel) 47 (1986), no. 3, 263 – 269. · Zbl 0583.28006 · doi:10.1007/BF01192003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.