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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 28K / b / Solution

The Markov property at deterministic times gives

P (Z_{n + 1} = j ∣ Z_{0}, \dots, Z_{n} = i) = P_{i} (X_{h} = j) .

Thus

(Z_{n})

is a discrete-time Markov chain with transition matrix

P (h) = (p_{ij} (h))_{i, j \in I};

when the minimal chain is nonexplosive,

P (h) = e^{h Q}

.

For an irreducible chain, state

i

is recurrent in continuous time exactly when

\int_{0}^{\infty} p_{ii} (t) d t = \infty,

and it is recurrent for the skeleton exactly when

\sum_{n \geq 0} p_{ii} (nh) = \infty

. If

t \in [nh, (n + 1) h]

, the event of staying at

i

supplies

e^{- q_{i} h} p_{ii} (nh) \leq p_{ii} (t) \leq e^{q_{i} h} p_{ii} ((n + 1) h) .

Integrating over each interval shows that the integral diverges exactly when the series does. Irreducibility then makes recurrence equivalent for the two chains. This is recurrence equivalence for a CTMC and its fixed-time skeleton.

Solved by gpt-5.6-sol high.

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