## Tuesday, April 10, 2007

### I completed the other half of the exercise now. So two exercises are now done.

I have now completed the second half of a problem concerning communicating classes of a markov chain transition probability matrix. What we do is this: "For state $i$ we evaluate $P_{ij}^{n}$ for all $j$. and any $n \ge 0$ and at the same time evaluate $P_{ji}^{m}$ for any $m \ge 0$. If we can show that $P_{ij}^{n} > 0$ for any $n \ge 0$ and also that $P_{ji}^{m} > 0$ for any $m \ge 0$ we have shown that the states $i$ and $j$ communicate. This allows us to establish which states communicate and then grouping all states by ability to communicate allows us to establish the different classes of states.