\infty$.

This shows that $\E X_n$ cannot be bounded in terms of $a, J$ and $V$, and by “stringing together” such examples, for larger and larger choices of $M$, we can clearly make $\sup_n \E(X_n) = \infty$. We conclude that condition (C1), combined with (C2) with $p=2$, still does not guarantee stability of $\{X_n\}$.

&gt;From the queueing theory perspective, it would be desirable, in condition (C2) of Theorem \[th:1\], to replace $|X_{n+1}-X_n|$ by $[X_{n+1}-X_n]^+$, i.e. to bound the $p^{\rm th}$ moments of just the [*positive part*]{} of the increments. Intuitively, this would correspond to allowing arbitrarily large [*negative*]{} increments, and bounding only the large [*positive*]{} increments. The problem with this is that the process is not sufficiently affected by its negative drift when this is all concentrated into a few unlikely large jumps. We give a counterexample to demonstrate this.

Fix $0<\epsilon<1$, and consider the following [*time-homogeneous*]{} Markov chain $\{X_n\}$. Let $X_0=0$, and for $n \ge 0$, let $$\P(X_{n+1} = 1 \, | \, X_n = 0) \ = \ 1$$ $$\P(X_{n+1} = x+1 \, | \, X_n = x > 0) \ = \ 1 - (1+\epsilon)/(x+1)$$ $$\P(X_{n+1} = 0 \, | \, X_n = x > 0) \ 