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Part BCSIR NET December 2025an-unreachable-absorbing-state-splits-the-chain-into-two-closed-classes

An unreachable absorbing state splits the chain into two closed classes

Let {Xn:n≥0X_{n} : n \ge 0} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix P=(pij)P = (p_{ij}) given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

  1. A.There is no stationary distribution
  2. B.Stationary distribution exists and is unique
  3. C.There are exactly two stationary distributions
  4. D.There are infinitely many stationary distributions✓

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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