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Part CCSIR NET December 2025reaching-a-closed-class-with-no-way-back-makes-every-state-outside-it-transient-not-just-the-one-adjacent-to-it

Reaching a closed class with no way back makes every state outside it transient not just the one adjacent to it

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

  1. A.State 2 is a positive recurrent state✓
  2. B.Mean recurrence time of state 1 is 7/4✓
  3. C.State 4 is a transient state✓
  4. D.State 3 is aperiodic and ergodic

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: An irreducible chain with a stationary distribution converges to it

More on this topic

The chapter behind this: Markov chains: classification and stationary behaviour — free to read

From Limit Theorems and Markov Chains › Markov chains: classification of states, stationary distributions

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