Question 11.2.31: In Example 11.11, define f(i) to be the proportion of G genes......
In Example 11.11, define f(i) to be the proportion of G genes in state i. Show that f is a harmonic function (see Exercise 27). Why does this show that the probability of being absorbed in state (GG,GG) is equal to the proportion of G genes in the starting state? (See Exercise 17.)
You can easily check that the proportion of G’s in the state provides a harmonic function. Then by Exercise 27 the proportion at the starting state is equal to the expected value of the proportion in the final aborbing state. But the proportion of 1s in the absorbing state GG,GG is 1. In the other absorbing state gg, gg it is 0. Thus the expected final proportion is just the probability of ending up in state GG,GG. Therefore, the probability of ending up in GG,GG is the proportion of G genes in the starting state.(See Exercise 17.)