论文标题
马尔可夫链的固定分布的凸表示的简短证明，并应用于状态空间截断
A Short Proof of a Convex Representation for Stationary Distributions of Markov Chains with an Application to State Space Truncation
论文作者
论文摘要
In an influential paper, Courtois and Semal (1984) establish that when $G$ is an irreducible substochastic matrix for which $\sum_{n=0}^{\infty}G^n <\infty$, then the stationary distribution of any stochastic matrix $P\ge G$ can be expressed as a convex combination of the normalized rows of $(I-G)^{-1} = \ sum_ {n = 0}^{\ infty} g^n $。在本说明中，我们简短地证明了这一结果，将理论扩展到了无数和连续的状态空间设置。该结果在获得涉及几乎可分解的马尔可夫链以及马尔可夫链的状态截断的算法中获得误差范围方面起着重要作用。我们还使用表示形式来建立针对截短的马尔可夫链绑定的新的总变化距离误差。
In an influential paper, Courtois and Semal (1984) establish that when $G$ is an irreducible substochastic matrix for which $\sum_{n=0}^{\infty}G^n <\infty$, then the stationary distribution of any stochastic matrix $P\ge G$ can be expressed as a convex combination of the normalized rows of $(I-G)^{-1} = \sum_{n=0}^{\infty} G^n$. In this note, we give a short proof of this result that extends the theory to the countably infinite and continuous state space settings. This result plays an important role in obtaining error bounds in algorithms involving nearly decomposable Markov chains, and also in state truncations for Markov chains. We also use the representation to establish a new total variation distance error bound for truncated Markov chains.
