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Let M be a homogeneous, discrete-time, finite-states Markov chain. When could we say that M is transient ?

  • When all states of M are transient ?

  • When M is not absorbing ?

I encountered both definitions when browsing MC documents, but they seem incompatible to me, unless there is something that I don't quite understand.

Could you enlighten me about the definition of a transient MC, if such a definition exists?

Andrew
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1 Answers1

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There is a theorem that says that if one state in communicating class is transient then all states in this class are transient. A communicating class is a set of states whose members communicate.

Then the definition of transient MC is as follows:
An irreducible MC is called transient if at least one (equivalently, every) state in this chain is transient. Irreducible chain is MC in which every state can be reached from every other state.

In other words you have to be able to reach each state from any state and if one state is transient then all of them are.

Hope that helps.

mfabi
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  • It raises (at least for me) an other question: is it possible to have not only one but all states transient even if the MC is not irreducible? – Andrew Jul 08 '22 at 20:37
  • Sorry for the gross mistake made in asking the question: in a Markov chain with finite state space, not all states can be transient together... End of the question! – Andrew Jul 10 '22 at 17:06