Transitory Set

Topics: State Set - Stochastic Process

---

(definition)

Let $T$ be a subset of state set $S$ and let $T^{\prime }$ be its complement in $S$.

If each state in $T$ can be reached from any other in $T$, and it’s possible to move from (at least) one state in $T$ to another in $T^{\prime }$, then we call $T$ a transitory set.

A transitory state is an element of a transitory set. A state that is not transitory is ergodic.

In simpler terms, a transitory state is a state where there exists a probability of getting out of it, but then never coming back (i.e. when reaching an ergodic set).

A Markov chain may have no transitory states. Compare how a Markov chain must have at least one ergodic set.