How do you show a Markov chain is transient?
Intuitively, transience attempts to capture how “connected” a state is to the entirety of the Markov chain. If there is a possibility of leaving the state and never returning, then the state is not very connected at all, so it is known as transient.
Are absorbing states recurrent?
You are correct: an absorbing state must be recurrent. To be precise with definitions: given a state space X and a Markov chain with transition matrix P defined on X. A state x∈X is absorbing if Pxx=1; neccessarily this implies that Pxy=0,y≠x.
Which is the theorem about recurrence and transience?
The next theorem shows that recurrence and transience are class properties: If some state in a communicating class is recurrent (resp. transient), then all states in this class are recurrent (resp. transient). Theorem 2.7.6.
What does transience and recurrence mean in a stochastic process?
Log in here. A stochastic process contains states that may be either transient or recurrent; transience and recurrence describe the likelihood of a process beginning in some state of returning to that particular state. There is some possibility (a nonzero probability) that a process beginning in a transient state will never return to that state.
Why is a strong form of recurrence necessary in a Markov chain?
The presence of many transient states may suggest that the Markov chain is absorbing, and a strong form of recurrence is necessary in an ergodic Markov chain. In a Markov chain, there is probability 111 of eventually (after some number of steps) returning to state xxx.
Is there a guarantee that a process will never return to a transient state?
There is some possibility (a nonzero probability) that a process beginning in a transient state will never return to that state. There is a guarantee that a process beginning in a recurrent state will return to that state.