## The Annals of Probability

### On the Existence of the Ergodic Hilbert Transform

R. Jajte

#### 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

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