  The classical theorem of Birkhoff states that the $T^N f(x) =
(1/N)\sum_{k=0}^{N-1} f(\sigma^k x)$ converges almost everywhere for $x\in X$
and $f\in L^{1}(X)$, where $\sigma$ is a measure preserving transformation of a
probability measure space $X$. It was shown that there are operators of the
form $T^N f(x)=(1/N)\sum_{k=0}^{N-1}f(\sigma^{n_k}x)$ for a subsequence
$\{n_k\}$ of the positive integers that converge in some $L^p$ spaces while
diverging in others. The topic of this talk will examine this phenomenon in the
class of Orlicz spaces $\{L{Log}^\beta L:\beta>0\}$.
