Transient State

A Transient State is a state that has a non-zero probability of never return to it.

• Context:
  • A state [math]\displaystyle{ s_i }[/math] of a Markov chain is called absorbing if it is impossible to leave it. A state which is not absorbing is called transient.
  • …
• Counter-Example(s):
  • Absorbing State

See: Markov Process State, Graph Path, Markov Chain, Accessible State, Recurrent State, Absorbing State.

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References

2011

• http://en.wikipedia.org/wiki/Markov_chain#Recurrence
  • QUOTE: A state i is said to be transient if, given that we start in state i, there is a non-zero probability that we will never return to i. Formally, let the random variable T_i be the first return time to state i (the "hitting time"): [math]\displaystyle{ T_i = \inf \{ n\ge1: X_n = i | X_0 = i\}. }[/math] The number [math]\displaystyle{ f_{ii}^{(n)} = \Pr(T_i = n) }[/math] is the probability that we return to state i for the first time after n steps. Therefore, state i is transient if [math]\displaystyle{ \Pr(T_i \lt {\infty}) = \sum_{n=1}^{\infty} f_{ii}^{(n)} \lt 1. }[/math]

    State i is recurrent (or persistent) if it is not transient. Recurrent states have finite hitting time with probability 1.

2010

• (Riddles, 2010) ⇒ (2010). “Introduction to Stochastic Processes.
  • QUOTE: Accessible: A state j is accessible from state i if there is a path from state i to state j. There is a path if pij(n) > 0 for some n > 0.

    Recurrent States: State i is recurrent if for every state j that is accessible from state [math]\displaystyle{ i }[/math], state i is also accessible from state j.

    Transient States: If a state is not recurrent, it is called transient.

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