50. To prove Equation (7.24), define the following notation:

Xj i ≡ time spent in state i on the j th visit to this state;

Ni(m) ≡ number of visits to state i in the first m transitions In terms of this notation, write expressions for

(a) the amount of time during the first m transitions that the process is in state i;

(b) the proportion of time during the first m transitions that the process is in state i.

Argue that, with probability 1,