(see e.g. Meyn and Tweedie, 1993), there are some connections. For example, it is known (see Tweedie, 1983, Theorem 2) that for $k\in\IN$, if $\{X_n\}$ is an aperiodic Harris-recurrent time-homogeneous Markov chain having stationary distribution $\pi(\cdot)$, and if $m_k \equiv
\int x^k \, \pi(dx) < \infty$, then for $\pi$-a.e. $x$, $\E_{\delta_x}
(X_n^k) \to m$, and hence $\sup_n \E_{\delta_x} (X_n^k) < \infty$. In other words, for such a Markov chain, stability in our sense is implied by standard Markov chain stability. In fact, it is known (e.g. Tuominen and Tweedie, 1994) that when $\{ X_n \}$ is a random walk with negative drift, reset to zero when it attempts to leave the nonnegative half-line and having square integrable increments, then $\E_{\delta_x} (X_n)$ will converge and hence be bounded. This shows that our (C1) and (C2) do represent a greater generality than the random walk context.

The paper is organized as follows. In Section 2 we state the main result, along with two extensions. (The extensions are reasonably straightforward, but we include them in order to provide readily referenceable results that don’t assume more than is needed.) We also provide in Section 2 a simpler proof of the main theorem in the 