Types of Markov chains and Markov states

According to the properties of the ordering of states, different sets of states may emerge [Kemeny SnellKemeny Snell1976]:

1. if it is possible to transit directly from any state $s_i$ to $s_j$, the set of states is called ergodic ; once entered in an ergodic set, it is never left;
2. an absorbing state is one where it is not possible to leave it once encountered ($p_{ii}=1$).
3. all other elements are called transient ; that is there is no possibility to stay in state $s_i$; once the system leaves a transient set, it never returns to it;

In every transitional probability matrix#3479#> there is an ergodic set, consisting of at least one element, but not necessarily a transient set. If a chain as more than one ergodic set without a transient set, it means that there are no interaction between the two sets.

If the chain is ergodic without transient sets, it may be regular , where after a sufficient lapse of time the system could be in any state. It may also be cyclic , where the system encounters a series of different states and return to the original states after $d$ steps.

If the chain has transient sets, the system will move to an ergodic set, either regular or cyclic; it cannot escape from an ergodic set once it enters one.