Abstract
Let $u$ be a unitary operator acting in $\mathbb{L}_2(\Omega, F, p)$, where $p$ is a probability measure. We prove that the limit $\lim_{n\rightarrow\infty}\sum_{0 < |k| \leq n} u^k f/k$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$ if and only if the limit $\lim_{n\rightarrow\infty} n^{-1}\sum^{n-1}_{k=0}u^kf$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$.
Citation
R. Jajte. "On the Existence of the Ergodic Hilbert Transform." Ann. Probab. 15 (2) 831 - 835, April, 1987. https://doi.org/10.1214/aop/1176992176
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