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Transient State
A state that a Markov chain has positive probability of leaving forever, so it is visited only finitely many times.
In Plain English
A transient state is one the chain might well pass through, but there's a real chance it never comes back once it leaves. Contrast this with a recurrent state, which the chain is guaranteed to revisit. Because there's some escape probability at every visit, a transient state ends up being visited only a finite number of times before the chain wanders off for good, a geometric-style countdown to the last visit.
Definition
A state is transient if the probability of eventually returning to , starting from , is strictly less than 1: . Equivalently, the number of visits to a transient state, starting from that state, follows a geometric distribution with a finite expected value, unlike a recurrent state, which is visited infinitely often. Every state in a finite-state Markov chain is either transient or recurrent (this dichotomy is exhaustive), and in an absorbing Markov chain the non-absorbing states are, by construction, exactly the transient ones.
Properties
- Number of visits to a transient state before the chain leaves it for good follows a geometric distribution, so 'transient' has a precise probabilistic signature, not just an informal sense of 'temporary.'
- In a finite-state chain, transient states cannot be revisited forever simply because there isn't room, eventually the chain must settle permanently into some recurrent class or absorbing state, since visits to transient states are almost surely finite in number.
- Classifying every state of a chain as transient or recurrent (via the transition matrix) is usually the first analytical step before computing anything about long-run behavior, since the entire question of whether a stationary distribution exists hinges on this classification.
At a Glance