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

Abstract

Let $T$ be a power-bounded operator on $L_p(\mu)$, $1< p < \infty$. We use a sublinear growth condition on the norms $\{\Vert \sum_{k=1}^n T^k f\Vert _p\}$ to obtain for $f$ the pointwise ergodic theorem with rate, as well as a.e.~convergence of the one-sided ergodic Hilbert transform. For $\mu$ 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 $L_p$. Combining them with results of Marcus and Pisier, we show that if $\{g_{n}\}$ is independent with zero expectation and uniformly bounded, then almost surely any realization $\{b_{n}\}$ has the property that for every $\gamma>3/4$, any contraction $T$ on $L_{2}(\mu)$ and $f \in L_{2}(\mu)$, the series $\sum_{k=1}^{\infty} \sfrac{b_{k} T^{k} f(x)}{k^{\gamma}}$ converges $\mu$-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of $L_{1}(\mu)$ of a probability space and $f \in L_{p}(\mu)$, $1<p< \infty$, the series $\sum_{k=1}^{\infty} \sfrac{b_{k} T^{k} f(x)}{k^{\gamma}}$ converges a.e. for $\gamma \in (\max\{\frac{3}{4},\frac{p+1}{2p}\},1]$.