The Banach-Saks p-property

We obtain general theorems which enable the calculation of the Banach-Saks index set for rearrangement invariant function spaces in terms of their Boyd indices. For many important classes of rearrangement invariant spaces, the index set is non-trivial if and only if the Boyd indices of the underlying space are non-trivial, although the latter equivalence does not hold in full generality. We also study Banach-Saks index sets defined for weakly null sequences of (identically distributed) independent random variables and their connections with the classical Banach-Saks index set.