Abstract: Let T be a positive positive linear Lp contraction, $1\leq p < \infty$. It is shown that norm convergence of the averages along a subsequence ${\bf n}$ for a positive Lp contraction can be obtained from the norm convergence of the averages along the same subsequence for operators induced by measure preserving transformations. This result is obtained in the more general setting of superadditive processes with respect to positive Lp contractions. In addition, the problem is investigated for two distinct definitions for the moving averages of superadditive processes.