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Part CCSIR NET June 2025a-2-cycle-and-a-3-cycle-in-the-same-class-force-the-period-to-1

A 2 cycle and a 3 cycle in the same class force the period to 1

Consider a Markov chain {Xn:n≥1X_{n} : n \ge 1} on state space {1, 2, 3, 4, 5} with the transition probability matrix whose rows are (0, 1/2, 1/2, 0, 0), (0, 0, 1, 0, 0), (0, 1/3, 0, 1/3, 1/3), (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1). Then, which of the following statements are true?

  1. A.Stationary distribution is (0, 0, 0, 0, 1).✓ Correct
  2. B.State 5 is absorbing and recurrent.✓ Correct
  3. C.All states are aperiodic.✓ Correct
  4. D.limn→∞p55lim_{n}\to\infty p_{55}⁽n^{n}⁾ = 1.✓ Correct

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

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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