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On randomly weighted one-sided ergodic Hilbert transforms

Published online by Cambridge University Press:  22 December 2004

C. CUNY
C. CUNY
Affiliation:
Ben Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel (e-mail: cuny@univ-nc.nc)
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Abstract

In this paper we obtain (almost) optimal results concerning randomly weighted one-sided ergodic Hilbert transforms. Given an iid sequence of centered random variables (Xn) in L log L, we show that there exists a universal set $\Omega'$ of probability 1 such that for any $\omega\in \Omega'$ the realization $(X_n(\omega))$ is good for the almost everywhere convergence of the weighted one-sided ergodic Hilbert transform associated with any dynamical system and any $g\in L\log L$. The method applies to powers along subsequences with ‘small’ growth and when considering Dunford–Schwartz operators instead of pointwise transformation. If the (Xn) are symmetric, but only in L log log L, we obtain a slightly weaker result.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 1 , February 2005 , pp. 89 - 99
Copyright
2004 Cambridge University Press

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