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Part CCSIR NET December 2024no-state-of-a-finite-irreducible-chain-is-transient

No state of a finite irreducible chain is transient

Consider the Markov chain with state space {0, 1, 2} and the transition probability matrix P = [[0, 1/2, 1/2], [3/4, 0, 1/4], [3/4, 1/4, 0]]. Let P⁽n^{n}⁾ = ((P⁽n^{n}⁾ij))_{ij})) denote the n-step transition probability matrix. Then, which of the following statements are true?

  1. A.P⁽2^{2}⁾00=3/4_{00} = 3/4✓ Correct
  2. B.P⁽3^{3}⁾10=39/64_{10} = 39/64✓ Correct
  3. C.The stationary probability that the chain is in state 2 is 2/7.✓ Correct
  4. D.State 1 is transient.

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, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

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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