Discrete-time markov chain (continuation)

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Discrete time Markov Chain

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Presentation transcript:

Discrete-time markov chain (continuation)

0.4 Definition State A 0.6 State C is accessible from state A State B is NOT accessible from state C 0.2 State B State C 0.8

0.4 Definition State A 0.6 State C is accessible from state A 𝒑 𝑨𝑪 (𝒏) >𝟎 for some 𝒏≥𝟎. 0.2 State B State C 0.8

Definition 0.4 0.6 State B is accessible from state B. 0.2 0.8 State A 𝒑 𝑩𝑩 (𝟎) = 𝑷 𝑿 𝟎 = 𝑩 𝑿 𝟎 =𝑩 =𝟏 0.2 State B State C 0.8

Definition 0.4 0.6 State A and B communicate. 0.2 0.8 State A State B is accessible from state A AND State A is accessible from state B State A and B communicate. 0.2 State B State C 0.8

Definition 0.4 0.6 0.2 0.8 Any state communicates with itself. State A 0.6 Any state communicates with itself. “Communicate” is transitive. 0.2 State B State C 0.8

0.4 Definition State A 0.6 If ALL states communicate, the Markov Chain is irreducible. 0.1 0.2 State C 0.9 State B 0.8

0.4 ClassificationS State A State B is transient iff there exists other states that is accessible from B but not vice versa. Once the process exits B, it will never return to B again. 0.6 0.5 State C 1 State B 0.5

0.4 ClassificationS State A State B is transient iff there exists other states that is accessible from B but not vice versa. A transient state will be visited only a finite number of times. 0.6 0.5 State C 1 State B 0.5

0.4 Classifications State A A state which is not transient is called recurrent. Once the process exits the recurrent state, the process will definitely return to this state again. 0.6 0.1 State C 0.9 0.5 State B 0.5

ClassificationS 0.4 0.6 0.5 0.5 State C is absorbing iff 𝒑 𝑪𝑪 =𝟏. State A State C is absorbing iff 𝒑 𝑪𝑪 =𝟏. Upon entering C, the process will never leave C. 0.6 0.5 State B State C 0.5

ClassificationS 0.4 0.6 0.5 1 0.5 State C is absorbing iff 𝒑 𝑪𝑪 =𝟏. State A State C is absorbing iff 𝒑 𝑪𝑪 =𝟏. Upon entering C, the process will never leave C. 0.6 0.5 State C 1 State B 0.5

Remark In a finite-state Markov Chain, at least one state is recurrent (i.e., not all states can be transient).

MONTE CARLO SIMULATION EXERCISE 0.6 MONTE CARLO SIMULATION EXERCISE State 1 0.4 0.3 0.9 State 3 0.1 State 2 0.7