$p>4$ it is straightforward to check (by integrating first) that it is finite. This gives the result. $\Cox$

We here present a few counterexamples to show that the hypotheses of Theorem 1 (in particular, the restriction that $p>2$) are really necessary.

The following example is due to Madhu Sudan (personal communication via A. Borodin). Let $\{X_n\}$ be a time-inhomogeneous Markov chain such that $\P(X_2=0) = 2/3 = 1 - \P(X_2=2)$, and such that for $n \ge 2$, $$\P(X_{n+1} = n+1 \, | \, X_n = n) = 1 - 2/n$$ $$\P(X_{n+1} = 0 \, | \, X_n = n) = 2/n$$ $$\P(X_{n+1} = n+1 \, | \, X_n = 0) = 1/n$$ $$\P(X_{n+1} = 0 \, | \, X_n = 0) = 1 - 1/n$$ These transition probabilities were chosen to ensure that $$X_n \ = \
\left\{
\eqalign{
0,& \quad {\rm prob} \ 2/3 \cr
n,& \quad {\rm prob} \ 1/3 }
\right.$$ for all $n \ge 2$. Hence, $\E(X_n) = n/3$, so that $\sup_n \E(X_n) = \infty$.

On the other hand, it is easily verified that (C1) is satisfied with $a=1$ and $J=0$. Furthermore, (C2) is satisfied with $p=1$ and $V=3$. We conclude that condition (C1) alone, or 