Let X1, X2,... be independent random variables such that P{X, = j} = a,,j0. Say that a record occurs at time n if X, > max(X,,...,xn1), where X, and if a record does occur at time n call X, the record value. Let R, denote the ith record value.

(a) Argue that {R,, i 1} is a Markov chain and compute its transi- tion probabilities.

(b) Let T, denote the time between the ith and (i + 1)st record. Is {T,, i 1} a Markov chain? What about {(R,, T), i 1}? Compute transition probabilities where appropriate.

(c) Let S = T., n 1. Argue that {S,, n 1} is a Markov chain 1-1 and find its transition probabilities.