Abstract.
The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L 1(ℝ/ℤ), then the averages
converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L 1(ℝ/ℤ), the averages A n aƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.
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Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000
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Rio, E. Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov . Probab Theory Relat Fields 118, 342–348 (2000). https://doi.org/10.1007/PL00008745
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DOI: https://doi.org/10.1007/PL00008745
Keywords
- Convergence Rate
- Obtain Convergence Rate