piece can be taken to be $\DE_n \one_{\DE_n \leq Z_n}$ for some adapted nonpositive $\{ Z_n \}$. In other words, the moment condition need not apply to the negative tail of the increment, as long as the mean is still nonpositive when the negative tail is excluded. We state this more precisely as the following corollary.

\[cor:better\] Let $\{ Y_n \}$ be adapted to $\{ \F_n \}$ with $Y_0 \leq 0$. Let $\{ Z_n \}$ be any adapted nonpositive sequence. Suppose $\E (|\DE_n|^p \one_{\DE_n > Z_n} \| \F_n) \leq B$ for all $n$ and $\E (\DE_n \one_{\DE_n > Z_n} \| \F_n) \leq 0$ for all $1 \leq n < \sigma$, where $\DE_n = Y_{n+1} - Y_n$ and $\sigma = \inf \{ n > 0 : Y_n \leq n \}$. Then for $0 < r < p$, $$\E \left( (Y_t^+)^r \one_{\sigma > t} \right) \leq K t^{r - p} .$$ $\Cox$

We now use these corollaries to derive Theorem \[th:1\] and Corollary \[th:9\].

[Proof of Theorem  from Corollary , and of Corollary \[th:9\] from Corollary ]{}: First assume that $a = 1$ and $J = 0$. Given $\{ X_n \}$ as in the hypotheses of the theorem, fix an $N \geq 