4. (Reflected random walk) Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 =...

4. (Reflected random walk) Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 = X0 + Zn+1 − Xn m=0 min{0, Xm + Vm+1 − Um+1}, where Zn = Xn m=1 (Vm − Um), n ≥ 1. Q5. (Extreme value process) Let {In|n ≥ 0} be an i.i.d. sequence of Z-valued random variables such that P{I1 = k} = pk, k ∈ Z and pk > 0 for some k > 0. Define Xn = max{I1, I2, · · · , In}, n ≥ 1, X0 = 0. Show that {Xn|n ≥ 0} is a Markov chain. Also find the transition matrix.

Q6. (Truncated input-output process) In a discrete time input-output system, let Xn denote the number of units of an item at end of the nth period, (n − 1, n]. In each period (n − 1, n], the system has an input(arrival) Vn and an output (departure) Un, n ≥ 1. The system disregard any output if its exit makes the state of the system negative. We assume that {(Un, Vn)|n ≥ 1} is an i.i.d. sequence such that Un is independent of Vn for all n and P{U1 = k} = qk, k ≥ 0 and P{V1 = k} = pk, k ≥ 0. Assume that X0 is Z +-valued random variable independent of {(Un, Vn)|n ≥ 1}. Show that {Xn|n ≥ 0} is a Markov chain

Q5. (Extreme value process) Let {In|n ≥ 0} be an i.i.d. sequence of Z-valued random variables such that P{I1 = k} = pk, k ∈ Z and pk > 0 for some k > 0. Define Xn = max{I1, I2, · · · , In}, n ≥ 1, X0 = 0. Show that {Xn|n ≥ 0} is a Markov chain. Also find the transition matrix.

Expert Solution

orchestra answered 1 year ago

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