Question in pic

Let X 1 , X 2, . . . be independent random variables such that P{X, = j} = a,,j 2 0. Say that a record occurs at time n if X" > max(X1, . . . , X?!) where X0 = 00, 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 a 1} is a Markov chain and compute its transi- tion probabilities. (b) Let T, denote the time between the ith and (1' + 1)st record. Is {Tn i 2 1} a Markov chain? What about {(R,, '11), t\" 2 1}? Compute transition probabilities where appropriate. (c) Let 5,, = 2 7",, n a 1. Argue that {3", n a 1} is a Markov chain :=I and nd its transition probabilities