Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 18H / a

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- Solution a

Solution

The chain is an irreducible birth-death process. The detailed balance equations are

π_{i} \frac{1}{3} = π_{i + 1} \frac{2}{3},

so

π_{i + 1} = π_{i} /2

. Normalization gives the stationary distribution

π_{i} = 2^{- (i + 1)}, i \geq 0 .

An irreducible countable-state Markov chain that possesses a stationary probability distribution is positive recurrent. Hence

X

is positive recurrent.

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