  Consider the normalized partial sums of a real-valued function $F$ of a
Markov chain, \[\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.\] The
chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $\mathsf {X}$,
with transition kernel $P$, and it is assumed that the Lyapunov drift condition
holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$, $W:\mathsf
{X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under these
assumptions, the following conclusions are obtained: 1. It is known that this
drift condition is equivalent to the existence of a unique invariant
distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large numbers
holds for any function $F$ dominated by $W$:
\[\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.\] 2. The lower error
probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n\ge1$,
satisfies a large deviation limit theorem when the function $F$ satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If $W$ is near-monotone, then
control-variates are constructed based on the Lyapunov function $V$, providing
a pair of estimators that together satisfy nontrivial large asymptotics for the
lower and upper error probabilities. In an application to simulation of queues
it is shown that exact large deviation asymptotics are possible even when the
estimator does not satisfy a central limit theorem.
