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GATE 2021 ST – Question 46

Stochastic Processes · Markov Chains · 2 marks · Multiple choice

A time-homogeneous discrete Markov chain has finite or countable state space S. Which statement is true?

  1. At least one recurrent state exists.
  2. An absorbing state implies at least one stationary distribution.
  3. If all states are positive recurrent,there is a unique stationary distribution.
  4. An irreducible chain on {1,2} always satisfies lim P(X_n=i | X₀=i)=π_i for a stationary π.

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Correct answer: (B) An absorbing state implies at least one stationary distribution.

Explanation

An absorbing state supports its own stationary point mass,proving **B**. A countable chain can drift deterministically to infinity with no recurrent state. Multiple closed positive recurrent classes give multiple stationary laws,refuting C. The deterministic two-state alternating chain is irreducible but periodic and lacks the asserted pointwise limit,refuting D.

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