Theorem]{} \[th:2\]: Let $T = \inf \{ k \geq 0 : 
\DD_k \geq t/4 \}$ be the time of the first large jump. Since $\tau > t$ implies $S_x < x$ for all $x \leq t$, we can write $$\label{eq:star}
\E \left( (M_t^+)^r \one_{\tau > t} \right) 
= \E \left( (M_t^+)^r \one_G \right) 
+ \E \left( (M_t^+)^r \one_H \right) ,$$ where $G = \{ T \geq S_{t/2} < t < \tau \}$ and $H = \{ T < S_{t/2} 
< t < \tau \}$.

To bound the first term, abbreviate $S := S_{t/2}$ and begin by observing that $M_S \leq 3t/4$ on $G$, since the level $t/2$ or higher has just been obtained and the increment was no more than $t/4$. Thus $$\E \left( (M_t^+)^r \one_G \right) 
\leq \P (\tau > S) \E ((M_t^+)^r \one_{\tau > t} \| \F_S) .$$ The first factor may be bounded via Lemma \[lem:tau\]: $$\label{eq:S}
\P (\tau > S) \leq {2^{p/2} C' \over t^{p/2}} \, .$$ The second factor is bounded using the formula $$\label{eq:parts}
\E \left( Z^r \one_{Z > u} \right) = u^r \P (Z > u) + \int_u^\infty r y^{r-1} 
   \P (Z > y) \, dy .$$ By Lemma \[lem:Lp\] conditionally on $\F_S$, 