Laws of large numbers with rates and the one-sided ergodic Hilbert transform

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

Let T be a power-bounded operator on Lp(μ), 1 < p < ∞. We use a sublinear growth condition on the norms {∥ k=1n Tk f∥p} to obtain for f the pointwise ergodic theorem with rate, as well as a.e. convergence of the one-sided ergodic Hilbert transform. For μ finite and T a positive contraction, we give a sufficient condition for the a.e. convergence of the "rotated one-sided Hilbert transform"; the result holds also for p = 1 when T is ergodic with T1 = 1. Our methods apply to norm-bounded sequences in Lp. Combining them with results of Marcus and Pisier, we show that if {gn} is independent with zero expectation and uniformly bounded, then almost surely any realization {bn} has the property that for every γ > 3/4, any contraction T on L2(μ) and f ∈ L2(μ), the series Σk=1 b kTk f(x)/kγ converges μ-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of L 1(μ) of a probability space and f ∈ Lp(μ), 1 < p < ∞, the series Σk=1 b kTk f(x)/kγ converges a.e. for γ ∈ (max{3/4, p+1/2p}, 1].

Original languageEnglish
Pages (from-to)997-1031
Number of pages35
JournalIllinois Journal of Mathematics
Volume47
Issue number4
DOIs
StatePublished - 1 Jan 2003

ASJC Scopus subject areas

  • Mathematics (all)

Fingerprint

Dive into the research topics of 'Laws of large numbers with rates and the one-sided ergodic Hilbert transform'. Together they form a unique fingerprint.

Cite this