## Long Run Proportions of Markov Chains

A Markov chain consisting of a selection of random choices between events A and B may be represented by the table below:

| Future State | |

Present State | ||

From this we may write down the transformation matrix

If the initial state of the system is represented by the vectorthen the state at timeis represented by

The state at timeis represented by

In generalĀ to state at timeis represented by

We can find the proportion of time in whch the system is in statesor by puttingWe obtain

Hence

From both of these equations

Also(since probabilities add to 1).

and similarly