problem about renewal reward process and wald's equation.

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 Xi , i > 1. (a) Let M1 = N. Argue that the sequence of random variables XN +1, XN1+2, . . . is independent of X1, ..., Xx and has the same distribution as the original sequence Xi, i > 1. Now treat XN +1, XN1+2, . . . as a new sequence, and define a stopping time N2 for this sequence that is defined exactly as N is on the original sequence. (For instance, if N1 = min(n: Xn > 0}, then N2 = min{n: XNitn > 0}.) Similarly, define a stopping time N3 on the sequence XN+N2+1, XNI+N2+2, .. . that is identically defined on this sequence as N is on the original sequence, and so on. (b) Is the reward process in which X; 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