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(Solved) 2. Let X1,X2, . .. be independent random variables such taht P(X,- = j) = 04,-, for 0 S j S J. Say that a record occurs at time n 2 1 if Xn > maX{X1,...


2. Let X1,X2, . .. be independent random variables such taht P(X,- = j) = 04,-, for 0 S j S J. Say that a record occurs at time n 2 1 if Xn > maX{X1, . . . ,X,,_1}, Where X0 2 —00, and if a record occurs at time n, call Xn the record value. Let R,- denote the i — th record value. (a) Argue that {R,- : i 2 1} is a Markov chain and compute its transition probabil— ities. Classify the states of the chain in transient, recurrent (positive or null), absorbing, etc. Is there a stationary distribution for the chain? Is there a limiting distribution? Are these unique? Consider two cases: J 2 00 and J < 00. (b) Let T,- denote the time between the ith and (i + 1)th record. Is {T,-,i 2 1} a Markov chain? What about {(R,,T,-),i 2 1}? Where appropriate, compute transition probabilities and classify the states of the chain.
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