to $\{ \F_n \}$ with $Y_0 \leq 0$. Suppose $\E (|\DE_n|^p \| \F_n) \leq B$ for all $n$ and $\E (\DE_n \| \F_n) \leq 0$ 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 $0 < r < p$ there is a constant $K = K (B , p , r)$ such that $$\E \left( (Y_t^+)^r \one_{\sigma > t} \right) \leq K t^{r - p} .$$

[Proof:]{} An easy fact useful here and later is that $z^+ \leq 1 + |z|^p$ and hence $$\label{eq:p to 1}
\E |Z|^p \leq b \; \Rightarrow \; \E Z^+ \leq 1 + b .$$ Recall (see e.g. Durrett 1996, p. 237) that the supermartingale $\{ Y_{n \wedge \sigma} : n \geq 1 \}$ may be decomposed as $Y_{n \wedge \sigma} = M_n - A_n$ where $\{ M_n : n \geq 1 \}$ is a martingale and $\{ A_n : n \geq 1 \}$ is an increasing predictable process with $A_1 = 0$. Let $\mu_n$ denote $\E (\DE_n \| \F_n)$. Then the increments $\DD_n := M_{n+1} - M_n$ satisfy $$\E (|\DD_n|^p \| \F_n) = \E (|\DE_n 