1$; we will compute an upper bound for $\E (X_N^+)^r$ that does not depend on $N$. Let $U := \max \{ k \leq N : X_k \leq 0 \}$ denote the last time up to $N$ that $X$ takes a nonpositive value. Decompose according to the value of $U$: $$\E (X_N^+)^r = \sum_{k=0}^{N-1} \E \left( (X_N^+)^r \one_{U = k} \right) .$$

To evaluate the summand, define for any $k < N$ a process $\{ Y^{(k)}_n \}$ by $Y^{(k)}_n = (X_{k + n} + n) \one_{X_k \leq 0}$. In other words, if $X_k > 0$ the process $\{ Y^{(k)}_N \}$ is constant at zero, and otherwise it is the process $\{ X_n \}$ shifted by $k$ and compensated by adding 1 each time step. We apply Corollary \[cor:1\] to the process $\{ Y^{(k)}_n \}$. Hypothesis (C1) of Theorem \[th:1\], together with the fact that $X_{k+j} > 0$ for $0 < j < \sigma^{(k)}$, imply that $\E (\DE_n \| \F_n) \leq 0$ when $1 \leq n \leq \sigma^{(k)}$. Also, $\E (|\DE_n|^p \| \F_n) \leq \E (|1 + X_{n+1} - X_n|^p \| \F_n) 
\leq B := 2^p (1 + V)$. The conclusion is that $$\E \left( [(Y^{(k)}_{N-k})^+]^r \one_{\sigma^{(k)} > N - k} \right) 