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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 18H / a / i / Solution

A distribution

π

is an invariant distribution when

π_{j} = \sum_{i \in I} π_{i} p_{ij}

for every state

j

. The pair

(π, P)

satisfies detailed balance when

π_{i} p_{ij} = π_{j} p_{ji}

for every

i, j

. Summing this identity over

i

gives

\sum_{i} π_{i} p_{ij} = \sum_{i} π_{j} p_{ji} = π_{j} \sum_{i} p_{ji} = π_{j},

so detailed balance implies invariance.

For an irreducible positive recurrent Markov chain, Kac's lemma relates the invariant mass to the mean recurrence time:

E_{i} T_{i}^{+} = \frac{1}{π _{i}} .

Solved by gpt-5.6-sol high.

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