模拟大函数的大偏差渐近性和控制变量
Large deviation asymptotics and control variates for simulating large functions
摘要：
考虑马尔可夫链的实值函数$F$的归一化部分和，\[ϕ_n:=n^{-1}\sum_{k=0}^{n-1}F(Φ(k)),\qquad n\ge1.\]链$\{Φ(k):k\ge0\}$在一般状态空间$\mathsf {X}$中取值，具有转移核$P$，并假设李雅普诺夫漂移条件成立：$PV\le V-W+b\mathbb{I}_C$其中$V:\mathsf {X}\to(0,\infty)$，$W:\mathsf {X}\to[1,\infty)$，集合$C$很小，$W$优于$F$。在这些假设下，本文得到了以下结论：1.已知该漂移条件等价于存在满足$π(W)<\infty$的唯一不变分布$π$，并且大数定律对于由$W$支配的任何函数$F$成立：\[ϕ_n\toϕ:=π(F),\qquad{a.s.}, n\to\infty.\] 2。当函数$F$满足单调性条件时，由$\mathsf {P}\{ϕ_n\le c\}$定义的下错误概率，对于$c<ϕ$，$n\ge1$，满足大偏差极限定理。在附加的小条件下，得到了精确的大偏差展开式。3.如果$W$是近单调的，那么控制变量的构造基于李雅普诺夫函数$V$，提供一对估计，一起满足非平凡的大渐近的上下误差概率。在应用程序模拟队列，它表明，确切的大偏差渐近是可能的，即使当估计不满足中心极限定理。
英文摘要：
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.
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