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

Expert Solution

milcah answered 11 months ago

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

Is the following statement always true? “If an irreducible Markov chain has period 2, then for...

Is the following statement always true? “If an irreducible Markov chain has period 2, then for every state i∈S we have P2ii>0.” (Prove if “yes”, provide a counterexample if “no”)

Stochastic Processes: 1. What does it mean for a Markov Chain to be irreducible? 2. What...

Stochastic Processes: 1. What does it mean for a Markov Chain to be irreducible? 2. What simple conditions imply that a Markov Chain is irreducible?

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 ⊂...

The following is the transition probability matrix of a Markov chain with states 1,2,3,4 ⎛⎞ .4...

The following is the transition probability matrix of a Markov chain with states 1,2,3,4 ⎛⎞ .4 .3 .2 .1 P=⎜.2 .2 .2 .4⎟ ⎝ .25 .25 .5 0 ⎠ .2 .1 .4 .3 (a) find the probability that state 3 is entered before state 4; If X0 = 1 (b) find the mean number of transitions until either state 3 or state 4 is entered.

A (time-homogeneous) Markov chain built on states A and B is depicted in the diagram below....

A (time-homogeneous) Markov chain built on states A and B is depicted in the diagram below. What is the probability that a process beginning on A will be on B after 2 moves? consider the Markov chain shown in Figure 11.14. Figure 11.14- A state transition diagram. Is this chain irreducible? Is this chain aperiodic? Find the stationary distribution for this chain. Is the stationary distribution a limiting distribution for the chain?

The following is the transition probability matrix of a Markov chain with states 1, 2, 3,...

The following is the transition probability matrix of a Markov chain with states 1, 2, 3, 4 P 0 1 2 3 0 .4 .3 .2 .1 1 .2 .2 .2 .4 2 .25 .25 .5 0 3 .2 .1 .4 .3 If Xnot = 1 (a) find the probability that state 3 is entered before state 4; (b) find the mean number of transitions until either state 3 or state 4 is entered.

Let Xn be a Markov chain with states 0,1,...,9 and transition probabilities P0,0 = P0,1 =...

Let Xn be a Markov chain with states 0,1,...,9 and transition probabilities P0,0 = P0,1 = P9,8 = P9,9 = 1/2 an Pi,i = Pi,i+1 = Pi,i−1 = 1/3 for all 1 ≤ i ≤ 8. (a) Draw the transition diagram. (b) What is the probability that X1,X2,X3,X4 are all smaller than 3 given that X0 = 1? Hint: Create a simpler Markov chain with 4 states.

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition...

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition matrix. 1/3 2/3 0 0 2/3 0 1/3 0 1/3 1/3 0 1/3 0 1/3 2/3 0 a.) Let X0 = 3 and let T3 be the first time that the Markov chain returns 3, compute P(T3 = 2 given X0=3). Please show all work and all steps. b.) Find the stationary distribution π. Please show all work and all steps.

Prove that the following is true for all positive integers n: n is odd if and...

Prove that the following is true for all positive integers n: n is odd if and only if n2 is odd.

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