What is the difference between “essential” and “recurrent”?
You can’t get out of an essential class,while in a recurrent class you know you have probability 1 of returning to your starting point. Hence ‘recurrent’ implies ‘essential’ – but not the other way round. See Theorems 2.33, 2.34 for exact statements, and look at Example 2.49 for a case where the two notions are not the same (in that example, the class {1,2,3,…} is essential, but is not recurrent if p> q). The two notions agree when the Markov chain state-space is finite. Unfortunately many interesting Markov chains in applications have infinite state-space (eg the integers).
