Abstract
No general statement can be made about the speed with which convergence takes place in the Ergodic Theorem, in the sense that one can never be sure that convergence of the remainder to zero is fast enough to make, through term-by-term multiplication, any divergent series become convergent. As a corollary we obtain the nonexistence in general of the one-sided ergodic Hilbert transform; that is, there is no pointwise “nonhomogeneous ergodic theorem”.
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Kakutani, S., Petersen, K. The speed of convergence in the Ergodic Theorem. Monatshefte für Mathematik 91, 11–18 (1981). https://doi.org/10.1007/BF01306954
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DOI: https://doi.org/10.1007/BF01306954