Moment conditions for a sequence with negative drift to be uniformly bounded in L^r

Suppose a sequence of random variables {X_n} has negative drift when above a certain threshold and has increments bounded in L^p. When p>2 this implies that EX_n is bounded above by a constant independent of n and the particular sequence {X_n}. When p=<2 there are counterexamples showing this does not hold. In general, increments bounded in L^p lead to a uniform L^r bound on X_n^+ for any r<p-1, but not for r>=p-1. These results are motivated by questions about stability of queueing networks.