Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 9C / i

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- Solution i

Solution

For a time-homogeneous Markov chain with transition matrix

P

, two states are communicating states when each can be reached from the other with positive probability in finitely many steps:

i \leftrightarrow j ⟺ \exists m, n \geq 0 : (P^{m})_{ij} > 0, (P^{n})_{ji} > 0.

The convention

P^{0} = I

includes communication of a state with itself.

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