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GATE 2020 ST – Question 52

Stochastic Processes · Markov Chains · 2 marks · Numerical answer

A Markov chain on{0,1,2} has $$P=\begin{pmatrix}0&1&0\\.3&0&.7\\0&1&0\end{pmatrix}.$$ Find $\lim_{n\to\infty}P(X_{2n}=2\mid X_0=2)$ (one decimal).

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Correct answer: 0.7 to 0.7

Explanation

From state2 the chain moves to1 with certainty. From1 it returns to2 with probability0.7 or to0 with0.3; both0 and2 next return to1. Thus at every positive even time, state2 has probability0.7 given the starting state2. The even-time limit is0.7 despite the chain’s period2.

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