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 $T1=1$.
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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