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

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 0sequence {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 Xn+ for any r<p−1, but not for r⩾p−1. These results are motivated by questions about stability of queueing networks.