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

Abstract. Let T be a power-bounded operator on Lp(µ), 1 < p <∞. We use a sublinear growth condition on the norms {‖∑nk=1 Tkf‖p} to obtain for f the pointwise ergodic theorem with rate, as well as a.e. con-vergence of the one-sided ergodic Hilbert transform. For µ finite and T a positive contraction, we give a sufficient condition for the a.e. con-vergence 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 indepen-dent with zero expectation and uniformly bounded, then almost surely any realization {bn} has the property that for every γ> 3/4, any con-traction T on L2(µ) and f ∈ L2(µ), the series k=1 bkT kf(x)/kγ con-verges µ-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of L1(µ) of a probability space and f ∈ Lp(µ), 1 < p <∞, the series k=1 bkT kf(x)/kγ converges a.e. for γ ∈ (max { 3 4, p+