case where $p > 4$ and $r = 1$, since in this case the back-of-the-napkin computation works, and anyone not interested in the sharp moment condition need read no further. In Section 3, we give examples to show why (C2) is needed with $p - 1 > r$ and why it is important to have moment bounds for the negative part of the increments as well as the positive part. Proofs are given in Sections 4 and 5, with Section 4 containing a reduction to a result on martingales and Section 5 containing a proof of the martingale result.

Throughout this paper, the filtration $\{ \F_n \}$ refers to any filtration to which $\{ X_n \}$ is adapted. We continue to use (C1) and (C2) for conditional expectations with respect to $\F_n$, slightly generalizing the notation of the introductory section.

\[th:1\] Let $X_n$ be random variables and suppose that there exist constants $a>0$, $J$, $V<\infty$, and $p > 2$, such that $X_0 \le J$, and for all $n$, $$\E (X_{n+1} - X_n \| \F_n) \leq -a \quad \mbox{ on the event }
\quad \{X_n > J\}
\eqno(\mbox{\rm C1})$$ and $$\E (|X_{n+1} - X_n|^p \| X_0 , \ldots , X_n) \leq 