IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
Which statement about a finite, irreducible, aperiodic Markov chain is correct?
A finite irreducible aperiodic chain has a unique stationary distribution, and the n-step transition probabilities converge to it whatever the starting state. Irreducibility gives uniqueness and aperiodicity gives convergence rather than cycling.
- AIt has no stationary distribution
- BIt has a unique stationary distribution, and n-step transition probabilities converge to it regardless of starting stateCorrect
- CIt has several stationary distributions, depending on the start state
- DIt has absorbing states
- Its n-step transition probabilities converge only if it starts in the stationary distribution
Explanation
A finite irreducible chain is positive recurrent and has a unique stationary distribution. Aperiodicity ensures the n-step probabilities converge to it from any initial state. Periodic chains would fail to converge, which is why option five is wrong.
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