24. Wald's equation can also be proved by using renewal reward processes. Let N be a stopping time for the sequence of independent and identically distributed random variables .

(a) Let Image. Argue that the sequence of random variables Image is independent of Image and has the same distribution as the original sequence .

Now treat Image as a new sequence, and define a stopping time Image for this sequence that is defined exactly as Image is on the original sequence. (For instance, if Image, then Image.) Similarly, define a stopping time Image on the sequence Image that is identically defined on this sequence as Image is on the original sequence, and so on.

(b) Is the reward process in which is the reward earned during period i a renewal reward process? If so, what is the length of the successive cycles?

(c) Derive an expression for the average reward per unit time.

(d) Use the strong law of large numbers to derive a second expression for the average reward per unit time.

(e) Conclude Wald's equation.