On the Existence of the Ergodic Hilbert Transform

Jajte, R.

Ann. Probab., Tome 15 (1987) no. 4, p. 831-835 / Harvested from Project Euclid

Résumé

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 $f \in \mathbb{L}_2(\Omega, F, p)$.

Détail

Publié le : 1987-04-14
Classification:  Ergodic Hilbert transform,  individual ergodic theorem,  spectral representation,  almost sure convergence,  47A35,  40A05