IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on states {A, B, C} has transition probabilities A→A 0.5, A→B 0.5, B→A 0.3, B→C 0.7, C→B 1. Which statement about the chain is correct?
The chain is irreducible because all three states communicate: A reaches C through B, and C reaches A through B. Hence there is a single class. State A is not absorbing and the finite irreducible chain has a unique stationary distribution.
- AThe chain is reducible because C cannot be reached from A
- BThe chain is irreducible, since every state can be reached from every other stateCorrect
- CState A is absorbing
- DState C is transient because it has only one exit
- The chain has no stationary distribution because it is not aperiodic
Explanation
A reaches B directly and C via B; B reaches A and C; C reaches B and then A. So all states communicate and the chain is irreducible. A has a self-loop with probability 0.5 so it is not absorbing, and the chain is aperiodic anyway. A finite irreducible chain has a unique stationary distribution.
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