The Annals of Probability

On the Existence of the Ergodic Hilbert Transform

R. Jajte

Full-text: Open access

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

Article information

Source
Ann. Probab., Volume 15, Number 2 (1987), 831-835.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992176

Digital Object Identifier
doi:10.1214/aop/1176992176

Mathematical Reviews number (MathSciNet)
MR885148

Zentralblatt MATH identifier
0634.47008

JSTOR
links.jstor.org

Subjects
Primary: 47A35: Ergodic theory [See also 28Dxx, 37Axx]
Secondary: 40A05: Convergence and divergence of series and sequences

Keywords
Ergodic Hilbert transform individual ergodic theorem spectral representation almost sure convergence

Citation

Jajte, R. On the Existence of the Ergodic Hilbert Transform. Ann. Probab. 15 (1987), no. 2, 831--835. doi:10.1214/aop/1176992176. https://projecteuclid.org/euclid.aop/1176992176


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