Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 27K / d

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- Solution d

Solution

Under the transience assumption, the number of returns to zero is geometrically distributed and hence finite almost surely. The final excursion is a pure ascent through all sufficiently large states. Its holding times are independent exponential random variables with means

\frac{1}{λ _{i} ( 1 + ρ _{i} )} .

The nonexplosion criterion for a pure birth process says that their sum is finite almost surely exactly when the sum of their means is finite. Thus

X is explosive ⟺ i = 1 \sum \infty \frac{1}{λ _{i} ( 1 + ρ _{i} )} < \infty.

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