manifested. This is the motivation for having stated Corollary \[th:9\] as an extension to the main theorem.

We will derive Theorem \[th:1\] and Corollary \[th:9\] from the following martingale result.

\[th:2\] Let $\{ M_n : n = 0 , 1 , 2 , \ldots \}$ be a sequence adapted to a filtration $\{ \F_n \}$ and let $\DD_n$ denote $M_{n+1} - M_n$. Suppose that the sequence started at $M_1$ is a martingale (i.e., $\E (\DD_n \| F_n) = 0$ for $n \geq 1$), and that $M_0 \leq 0$. Suppose further that for some $p > 2$ and $b > 0$ we have $$\label{eq:moment}
\E (|\DD_n|^p \| \F_n) \leq b$$ for all $n$ including $n=0$. Let $\tau = \inf \{ n > 0 : M_n \leq n \}$. Then for any $r \in (0 , p)$ there is a constant $C = C (b , p , r)$ such that $$\label{eq:conclusion}
\E \left( (M_t^+)^r \one_{\tau > t} \right) \leq C t^{r - p} .$$

We defer the proof of Theorem \[th:2\] until the following section. In the remainder of this section, we assume Theorem \[th:2\], and derive Theorem \[th:1\] and Corollary \[th:9\] as consequences.

\[cor:1\] Let $\{ Y_n \}$ be adapted 