assume any special properties of the increments such as independence, $r$-dependence, Markov property, symmetry, discreteness or nondiscreteness. Under what conditions can we guarantee that $\sup_n \E(X_n) < \infty$?

Specifically, we suppose that for some $a>0$ and some $J$ we have $$\E (X_{n+1} - X_n \| X_0 , \ldots , X_n) \leq -a \quad \mbox{ on the event }
\quad \{X_n > J\}
\eqno(\mbox{C1})$$ for all $n$. That is, the process has [*negative drift*]{} whenever it is above the point $J$. This condition alone says nothing about possible large jumps out of the interval $(-\infty , J)$, so we also assume that for some $p \ge 1$ and some $V<\infty$ we have $$\E (|X_{n+1} - X_n|^p \| X_0 , \ldots , X_n) \leq V
\eqno(\mbox{C2})$$ for all $n$. That is, the process has [*increments with bounded $p^{\rm th}$ moments*]{}. Conditions (C1) and (C2) are meant to characterize the behavior of sequences attracted to a basin, which always decreases in expectation except when it is already small. (One such example is a nonnegative Lyapunov function of a Markov chain; see for example Meyn and Tweedie, 1993.) Do these two conditions together imply that $\sup_n \E(X_n) < \infty$?

When $p=1$ the answer is 