Summary
Let (Ω, ⌆, P) be a probability space and let T be a measurable and measure preserving point transformation from Ω into Ω. Let f be a measurable and square integrable function on (Ω, ⌆, P), and let a N,k for N, K=0, 1, ... be such that \(\sum\limits_k {a_{N,k} = 1}\) for all N. The authors investigate conditions on the a N,k 's such that the sequence \(\sum\limits_{k = 0}^\infty {a_{N,k} f(T^k )}\) converges in mean square for all (Ω, ⌆, P, T) and f described above. The special cases T weakly mixing and T strongly mixing are also considered.
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Research partially sponsored by the Air Force Office of scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.
Research done while this author held a National Science Foundation Traineeship at the University of Missouri, Columbia.
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Hanson, D.L., Pledger, G. On the mean ergodic theorem for weighted averages. Z. Wahrscheinlichkeitstheorie verw Gebiete 13, 141–149 (1969). https://doi.org/10.1007/BF00537020
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DOI: https://doi.org/10.1007/BF00537020