no. Although an honest supermartingale has $\E X_n \leq \E X_0$ for all $n$, a process behaving like a supermartingale above $J$ may have $\E X_n$ unbounded. More surprisingly, the answer is still no for $p = 2$. However, if one assumes (C1) and (C2) with $p > 2$, then necessarily $\sup_n \E X_n < \infty$. More generally, we show (Theorem \[th:1\]) that (C1) and (C2) imply that $\sup_n \E\left(
(X_n^+)^r \right) < \infty$ whenever $r < p-1$, and that this bound on $r$ in terms of $p$ (or $p$ in terms of $r$) is sharp. Furthermore, our bounds are may be explicitly computed and depend only on the parameters $a$, $J$, $p$, $V$, and $r$.

Our results are motivated by questions about queueing networks. Specifically, several authors (Borodin et al., 1996, 1998; Andrews et al., 1996) consider network loads under the influence of a [*stochastic adversary*]{}. Here $X_n$ is the load of the network at time $n$. The adversary may add new packets to the network in virtually any manner, subject only to a load condition which leads to (C1) plus a moment condition such as (C2). (The load condition corresponds to the statement that, once the network is 