Recurrence and transience criteria

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Recurrence and transience criteria

We will study a second classification of states depending on the type of behavior of the chain (the criteria of recurrence and transience).

Let x be a state of the chain, we denote the time to reach x, denoted by Tx, the first instant when x is visited after departure. by convention, the reach time is infinite if we never reach x. The formula is as follows (we will use the classical notations for the probabilities):

recurrence and transience

If the chain starts from state x, we use the term return time.

A state x is said to be recurrent if:recurrence and transience

State x is said to be transient or transient otherwise, i.e. when:

recurrence and transience

 

A state is recurrent if we are sure to return to it, it is transient if there is a non-zero probability of never coming back to it, and therefore of leaving it definitively.

An equivalence class is said to be recurrent, respectively transient, if one of its vertices is recurrent, resp. transient.

A recurring class is closed, in other words, the probability of exiting a recurring class is zero.

Let x be any state belonging to the recurrence class C. Suppose
that there exists y ∉ C such that x → y and let us show that we have a contradiction. Note first that y does not lead to any vertex of C, because otherwise we would have y → x and therefore x ↔ y and y ∈ C. Moreover, we have:

recurrence and transience

However, the probability of not going back to x is bounded inferiorly by the probability of going to y in finite time (since y does not lead to any state of C). So, we have the following relation:

recurrence and transience

Which is a contradiction with recurrent x. We see that a recurring class is closed, but the converse is false in general, it is still verified if this class has a finite cardinality.

It is important to retain the following corollary: markov chain defined on a finite state space admits at least one recurrent state.
 
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