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

Guy Cohen, Michael Lin
2003 Illinois Journal of Mathematics  
Let T be a power-bounded operator on Lp(µ), 1 < p < ∞. We use a sublinear growth condition on the norms { n k=1 T k 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 T 1 = 1. Our methods apply to norm-bounded
more » ... es 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 L 2 (µ) and f ∈ L 2 (µ), the series ∞ k=1 b k T k f (x)/k γ 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 ∞ k=1 b k T k f (x)/k γ converges a.e. for γ ∈ (max{ 3 4 , p+1 2p }, 1].
doi:10.1215/ijm/1258138088 fatcat:dmrpc37uzraylpcpttzkzyjc4m