论文标题
与相互作用的加强随机步行近似准平台分布
Approximating Quasi-Stationary Distributions with Interacting Reinforced Random Walks
论文作者
论文摘要
我们提出了两个数值方案，用于近似具有吸收状态的有限状态马尔可夫链的准平台分布（QSD）。这两种方案都是用某些相互作用的链来描述的，在这些链中，通过系统中所有粒子的总时间占用度量给出了相互作用，并且以适当的方式加强过渡到收集粒子花费更多时间的状态的影响。这些方案可以看作是结合了基于QSD的两种基本模拟方法的关键特征，该方法分别源自Fleming and Viot（1979）和Aldous，Flannery和Palacios（1998）。这里研究的两个方案之间的关键区别在于，在第一种方法中，一个$ 0 $ $ 0 $的$ a（n）$粒子开始，粒子数量随时间持续保持恒定，而在第二种方法中，我们从一个粒子开始，每次以每种粒子的速度添加一个粒子，以这种方式以$ n $ n $ n $ $ n $ $ n $ $ a（n）$粒子的方式添加。我们几乎可以肯定地融合了唯一的QSD，并根据$ a（n）= o（n）$的关键假设建立了两个方案的中心限制定理。当$ a（n）\ sim n $时，预计波动行为将是非标准的。提出了一些探索性数值结果，以说明两个近似方案的性能。
We propose two numerical schemes for approximating quasi-stationary distributions (QSD) of finite state Markov chains with absorbing states. Both schemes are described in terms of certain interacting chains in which the interaction is given in terms of the total time occupation measure of all particles in the system and has the impact of reinforcing transitions, in an appropriate fashion, to states where the collection of particles has spent more time. The schemes can be viewed as combining the key features of the two basic simulation-based methods for approximating QSD originating from the works of Fleming and Viot (1979) and Aldous, Flannery and Palacios (1998), respectively. The key difference between the two schemes studied here is that in the first method one starts with $a(n)$ particles at time $0$ and number of particles stays constant over time whereas in the second method we start with one particle and at most one particle is added at each time instant in such a manner that there are $a(n)$ particles at time $n$. We prove almost sure convergence to the unique QSD and establish Central Limit Theorems for the two schemes under the key assumption that $a(n)=o(n)$. When $a(n)\sim n$, the fluctuation behavior is expected to be non-standard. Some exploratory numerical results are presented to illustrate the performance of the two approximation schemes.
