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 N1 = N. Argue that the sequence of random variables XN1+1, XN1+2,... is independent of X1,...,XN and has the same distribution as the original sequence Xi,i 1.

Now treat XN1+1,XN1+2,... as a new sequence, and define a stopping time N2 for this sequence that is defined exactly as is N1 is on the original sequence.

(For instance, if N1 = min{n: Xn > 0}, then N2 = min{n: XN1+n > 0}.) Similarly, define a stopping time N3 on the sequence XN1+N2+1,XN1+N2+2,... that is identically defined on this sequence as is N1 on the original sequence, and so on.