IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A finite, irreducible, aperiodic Markov chain has stationary distribution π. Which statement is correct?
For a finite irreducible aperiodic chain, the n-step transition probabilities converge to the unique stationary probability π_j whatever the starting state. The limit is not generally uniform, and the chain need not start in stationarity.
- AThe n-step probabilities p_ij(n) converge to π_j regardless of the starting state iCorrect
- BThe chain must be in its stationary distribution at time 0
- CThe stationary distribution is not unique
- DAll states are transient
- The n-step probabilities converge to 1/(number of states) for every such chain
Explanation
For a finite irreducible aperiodic chain, the limit of p_ij(n) equals π_j independent of i. The limit is uniform only when π is uniform, such as for doubly stochastic matrices. States are positive recurrent, not transient, and π is unique.
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