无跳马尔可夫链稳定性的ODE方法及其在MCMC中的应用
The ODE method for stability of skip-free Markov chains with applications to MCMC
摘要：
在过去的十年中，流体极限技术已经成为分析排队网络的核心工具，并应用于性能分析、模拟和优化。在本文中，这些技术的一些推广到一般类的无跳马尔可夫链。在排队模型的情况下，流体近似是通过缩放时间，空间和初始条件由一个大的常数。所得到的流体极限是状态空间的“大部分”中的常微分方程（ODE）的解。随机模型的稳定性和更精细的遍历性质，然后从流体极限的稳定性。此外，类似于流体模型通常用于设计控制策略的嵌入式上下文，在此一般设置中的限制ODE的结构提供了对马尔可夫链的动力学的理解。这些结果通过应用到马尔可夫链蒙特卡罗方法来说明。
英文摘要：
Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization. In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in ``most'' of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.
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