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Convergence for moving averages. (English) Zbl 0674.60035

Assume T is an ergodic measure preserving point transformation from a probability space onto itself. Let \(\{(n_ k,\ell_ k)\}^{\infty}_{k=1}\) be a sequence of pairs of positive integers, and define the sequence of averaging operators \(A_ kf(x)=\ell_ k^{- 1}\sum^{\ell_ k-1}_{j=0}f(T^{n_ k+j}x)\). Necessary and sufficient conditions are given for this sequence of averages to converge almost everywhere. Weighted versions are also considered.

MSC:

60G10 Stationary stochastic processes
60F15 Strong limit theorems
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[1] DOI: 10.1073/pnas.59.2.349 · Zbl 0185.21806 · doi:10.1073/pnas.59.2.349
[2] DOI: 10.2307/2000442 · Zbl 0619.47004 · doi:10.2307/2000442
[3] Bellow, Contemp. Math. 28 pp 49– (1984) · Zbl 0587.28013 · doi:10.1090/conm/026/737387
[4] DOI: 10.2307/2039829 · Zbl 0278.28011 · doi:10.2307/2039829
[5] Zygmund, Trigonometric Series II (1968)
[6] Sueiro, Math. Proc. Camb. Phil 102 pp 131– (1987)
[7] de Guzman, Real Variable Methods in Fourier Analysis 46 (1988) · Zbl 0449.42001
[8] DOI: 10.2307/1970516 · Zbl 0186.20503 · doi:10.2307/1970516
[9] Schwartz, Proc. Amer. Math. Soc. none pp none– (none)
[10] DOI: 10.1007/BF01192003 · Zbl 0583.28006 · doi:10.1007/BF01192003
[11] DOI: 10.1016/0001-8708(84)90038-0 · Zbl 0546.42017 · doi:10.1016/0001-8708(84)90038-0
[12] del Junco, Math. 247 pp 185– (1979)
[13] DOI: 10.2307/1970308 · Zbl 0103.08903 · doi:10.2307/1970308
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