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

Suppose a sequence of random variables {Xn} has negative drift when above a certain threshold and has increments bounded in Lp. When p > 2 this implies that EXn is bounded above by a constant independent of n and the particular sequence {Xn}. When p ≤ 2 there are counterexamples showing this does not hold. In general, increments bounded in Lp lead to a uniform Lr 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.