why are the recurrent classes closed?

Question

i've recently started studying about markov chain, we call a communication class a recurrent one in a markov chain if by starting from that class we infinitely return to it with probability 1,with this in mind ,why recurrent classes are closed?it means why we can't leave a recurrent class if we start from it?

thanks

Answer 1

If a class is not closed, then it is transient and hence not recurrent. Don't forget that the classes involve all states that communicate with each other. Thus, they form a "sub" markov chain within the full markov chain.

Answer 2

Look, they are closed because of the following.

You are allowed to wander as long as you want in your Markov Chain. And you can start wandering beginning from any state.

There are two possibilities now:
a) you just started in a Markov chain from which you can always wander from every state to every other state, in that case, the Markov chain is closed
b) you started in a Markov chain in which there are regions that, once you entered that region, you can never get out of that region anymore (because there are no outgoing arrows of that region), every such region is called a closed class.

Now there are again two possibilities.
b1) you started in a closed region, in that case, you will keep revisiting (recurring) to the same states over and over again.
b2) you started in a non-closed region but if you wander around long enough, eventually you will fall in a region from which you can't get out anymore. And in this case you will keep revisiting (recurring) to all the states over and over again.

So in any way, a recurrent state is always in a closed class. :)