Prove that for a Markov chain on a finite state space, no states are null recurrent.

Expert Solution

milcah answered 4 months ago

For an irreducible Markov chain, either all states are positive recurrent or none are. Prove.

Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · ·...

Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ) given by pij = 1 2 if j = i − 1 1 2 if j = i + 1, i ≥ 1, and p00 = p01 = 1 2 . Find P{X0 ≤ X1 ≤ · · · ≤ Xn|X0 = i}, i ≥ 0 . Q2. Consider the Markov chain given in Q1. Find P{X1,...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S. For i ∈...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S. For i ∈ S, define τi = min{n ≥ 0|Xn = i}. Show that τi is a stopping time for each i. Q2. Let τi as in Q1. but for any discrete time stochastic process. Is τi a stopping time? Q3. Let {Xn|n ≥ 0} be a Markov chain and i is a state. Define the random time τ = min{n ≥ 0|Xn+1 = i}. If τ...

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...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4 0.2 0.4 P = 0.6 0.3 0.1 0.5 0.3 0.2 And initial probability vector a = [0.2, 0.3, 0.5] For the Markov Chain with state-space, initial vector, and transition matrix discuss how we would calculate the follow; explain in words how to calculate the question below. a) P(X1 = 0, X2 = 0, X3 = 1, X4 = 2|X0 = 2) b) P(X2 =...

1.14 Let Xn be a Markov chain on state space {1,2,3,4,5} with transition matrix P= 0...

1.14 Let Xn be a Markov chain on state space {1,2,3,4,5} with transition matrix P= 0 1/2 1/2 0 0 0 0 0 1/5 4/5 0 0 0 2/5 3/5 1 0 0 0 0 1/2 0 0 0 1/2 (a) Is this chain irreducible? Is it aperiodic? (b) Find the stationary probability vector.

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7...

Xn is a discrete-time Markov chain with state-space {1,2,3}, transition matrix, P = .2 .1 .7 .3 .3 .4 .6 .3 .1 a) find E[X1|X0=2] = b) The P(X9=1|X7=3) = C) The P(X2=2) =

Q4. Prove or disprove that τ is a stopping time (with respect to the Markov chain...

Q4. Prove or disprove that τ is a stopping time (with respect to the Markov chain {Xn|n ≥ 0} iff {τ > n} ∈ σ(X0, · · · , Xn), ∀ n ≥ 0 Q5. Prove or disprove that τ is a stopping time (with respect to the Markov chain {Xn|n ≥ 0}) iff {τ ≥ n} ∈ σ(X0, · · · , Xn), ∀ n ≥ 0 Q6. Let {Xn|n ≥ 0} be a Markov chain and A ⊂...

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