finite for $\real(w) > 1$.) Then for general $a$ and $J$, we have $c(p,a,V,J,r) = J + a^r c(p,1,V/a^p,0,r)$. Now, these formulae are clearly rather messy, and may be of limited practical use. However, it may still be helpful to have them available for ready reference.**

We also state an extension allowing the negative part of the increments to avoid the moment condition in (C2):

\[th:9\] The conclusion of Theorem \[th:1\] still holds when $X_{n+1} - X_n$ is replaced by $(X_{n+1} - X_n) \one_{X_{n+1} - X_n > Z_n}$ in conditions (C1) and (C2), and $Z_n \leq -a$ is any sequence adapted to $\{ \F_n \}$.

The proof of Theorem \[th:1\] proceeds by decomposing according to the last time $U$ before time $n$ that $\{ X_k \}$ was less than $J$. When $p > 4$, Markov’s inequality, together with a crude $L^p$ estimate on $X_n - X_U$, gives bounds on the tails of $X_n$ sufficient to yield Theorem \[th:1\]. We finish the section by giving this argument.

Assume the notation and hypotheses of Theorem 1. Fix a positive integer $n$. Let $U = \max\{ k \le n; \ X_k \le J \}$. Let $\mu_i = \E\left( X_{i+1}-X_i \, | \, \F_i 