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2. Let {X t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₁ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = j, Xo = io, X₁ = 1,..., Xn-1=in-1 Xn = i) =P(Xn+1 =jXn = i)P(Xo = io, X₁ = ₁,..., Xn-1=in-1|Xn = i) for any j, io, i1,..., in-1,i in the state space.

2. Let {X t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₁ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = j, Xo = io, X₁ = 1,..., Xn-1=in-1 Xn = i) =P(Xn+1 =jXn = i)P(Xo = io, X₁ = ₁,..., Xn-1=in-1|Xn = i) for any j, io, i1,..., in-1,i in the state space.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 13EQ
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2. Let {Xt: t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₂ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = J, Xo = 20, X₁ = 1,..., Xn-1 = in-1|Xn = i) = P(Xn+1 =jXn = i) P(Xo = io, X1 = 1,..., Xn-1 = in-1 Xn = i) for any j, io, i1,..., in-1,i in the state space.
Transcribed Image Text:2. Let {Xt: t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₂ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = J, Xo = 20, X₁ = 1,..., Xn-1 = in-1|Xn = i) = P(Xn+1 =jXn = i) P(Xo = io, X1 = 1,..., Xn-1 = in-1 Xn = i) for any j, io, i1,..., in-1,i in the state space.
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