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.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as...

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as a random walk on a group (G, ·) with transition probabilities given by pxy = µ(y · x −1 ) for an appropriate distribution µ on G. Q2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn = 1}Un+1 + I{Xn 6= 1}Vn+1, n ≥ 0, where {(Un, Vn)|n ≥ 1} is an i.i.d. sequence of random variables such...

Let Xn is a simple random walk (p = 1/2) on {0, 1, · · ·...

Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing boundaries. Suppose X0 = 50. Let T = min{j : Xj = 0 or N}. Let Fn denote the information contained in X1, · · · , Xn. (1) Verify that Xn is a martingale. (2) Find P(XT = 100). (3) Let Mn = X2 n − n. Verify that Mn is also a martingale. (4) It is known that...

Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is...

Let x1 > 1 and xn+1 := 2−1/xn for n ∈ N. Show that xn is bounded and monotone. Find the limit. Prove by induction

Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U...

Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U ∈ Rn×n be an orthogonal matrix (UTU = UUT = In) and nd the distribution of UTx. Let y = (y1,...,yn) ∼ N(0,Σ) be a MVN random vector in Rn. Let Σ = UΛUT be the spectral decomposition of Σ. (b) Someone claims that the diagonal elements of Λ are nonnegative. Is that true? (c) Let z = UTy and nd the distribution of...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine a) the MLE ? of ?. b) E(? ̂). c) the asymptotic variance of the MLE of ?. d) the MLE of SD(Xi ) = √ ?.

Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,...

Q5. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. 1 2 Q6. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition...

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2,...

Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, and transition probability matrix (pij ) given by   2 3 1 3 0 0 1 3 2 3 0 0 0 1 4 1 4 1 2 0 0 1 2 1 2   Determine all recurrent states. Q3. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2} and transition probability matrix (pij...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix (pij ). Let τi = min{n ≥ 1 : Xn = i}, i = 0, 1, 2, 3. Define Bij = {Xτj = i}. Is Bij ∈ σ(X0, · · · , Xτj ) ? Q2. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, X0 = 0, and transition...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is...

Let {xn} be a real summable sequence with xn ≥ 0 eventually. Prove that √(Xn*Xn+1) is summable.

Let X1, X2, . . . , Xn be a random sample of size n from...

Let X1, X2, . . . , Xn be a random sample of size n from a Poisson distribution with unknown mean µ. It is desired to test the following hypotheses H0 : µ = µ0 versus H1 : µ not equal to µ0 where µ0 > 0 is a given constant. Derive the likelihood ratio test statistic

Subjects