Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 7H / b

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- Solution b

Solution

In a transient irreducible chain, every state

j

is transient, so the expected number of visits satisfies

\sum_{n = 0}^{\infty} P^{n} (i, j) = E_{i} V_{j} < \infty.

The terms are nonnegative, and every convergent series has terms tending to zero. Therefore

P^{n} (i, j) \to 0 .

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