What is an example of Markov chain with no closed communicating class?
Closed class means that once we are in that class, there would be no escape from it.
I am thinking that an example would be the 1d random walk. But the problem is that the Markov chain in 1d random walk would be consisted a single class.
The second example that I can think of is $0\Rightarrow1\Rightarrow2\Rightarrow3\Rightarrow4\dots\Rightarrow1000\Rightarrow\dots\Rightarrow\infty$. In this example, every state is a class and that there will be no return.
What do you guys think?