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[**Moment conditions for a sequence with negative drift\
to be uniformly bounded in $L^r$**]{}\

Robin Pemantle[^1] and Je[f]{}frey S. Rosenthal[^2]

(August 27, 1998; revised January 12, 1999.)

[**ABSTRACT:**]{} 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 $\E X_n$ is bounded above by a constant independent of $n$ and the particular sequence $\{ X_n \}$. When $p \leq
2$ 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 $r \ge p-1$. These results are motivated by questions about stability of queueing networks.

[Keywords:]{} $L^p$, $p^{th}$ moments, supermartingale, martingale, linear boundary, Lyapunov function, stochastic adversary, queueing networks.

[Subject classification: ]{} Primary: 60G07; Secondary: 60F25.

Let $X_0,X_1,\ldots$ be a sequence of real-valued random variables. We wish to find a condition, along the lines of behaving like a supermartingale when sufficiently large, that will guarantee $\sup_n \E X_n < \infty$. In particular, we do not wish to 