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

Xj i

≡ time spent in state i on the jth 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,

(e) Combine parts (a), (b), (c), and

(d) to prove Equation (7.24).

Transcribed Image Text:

N(PPI) X (c) flj as m j=1 N,(m) d) Ni(m)/mi as m o.