Large deviation asymptotics and control variates for simulating large functions

Abstract

Consider the normalized partial sums of a real-valued function $F$ of a Markov chain, \[ϕ_n:=n^{-1}\sum_{k=0}^{n-1}F(Φ(k)),\qquad n\ge1.\] The chain $\{Φ(k):k\ge0\}$ 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 $π$ satisfying $π(W)<\infty$, and the law of large numbers holds for any function $F$ dominated by $W$: \[ϕ_n\toϕ:=π(F),\qquad{a.s.}, n\to\infty.\] 2. The lower error probability defined by $\mathsf {P}\{ϕ_n\le c\}$, for $c<ϕ$, $n\ge1$, 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.