Long-Run Behavior of Markov Processes

我們討論Markov Process的長時間行為時，通常會有三個議題：

1. Limiting distributions
2. Stationary distributions
3. Ergodicity

首先，limiting distribution指的是P(transition probabilities)隨著時間增加會趨近於固定，即P*P*P*...P=常數。

第二，stationary distribution指的是π(probabitlty in state)隨著時間增加會趨近於固定，即π=常數，而π=π*P。

要注意的是，limiting distribution和stationary distribution指的涵義是不相同的。

第三，Ergodicity的說明，先看看書上的意思(Fundamentals of Queueing Theory,4th Edition)：Closely associated with the concepts of limiting and stationary distributions is the idea of regodicity, which has to do with the information contained in one infinitely lon sample path of a process.