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Strong Markov property. Let X be a Markov chain on S, and let T be a random variable taking values in {0, 1, 2,...} with the property that the indicator function I{T=n), of the event that T = n, is a function of the variables X₁, X2, ..., Xn. Such a random variable T is called a stopping time, and the above definition requires that it is decidable whether or not T = n with a knowledge only of the past and present, Xo, X₁,..., Xn, and with no further information about the future. Show that P(XT+m=j | Xk = xk for 0 ≤ k < T, XT = i) = P(XT+m = j | XT = i) for m≥ 0, i, je S, and all sequences (xx) of states.

Strong Markov property. Let X be a Markov chain on S, and let T be a random variable taking values in {0, 1, 2,...} with the property that the indicator function I{T=n), of the event that T = n, is a function of the variables X₁, X2, ..., Xn. Such a random variable T is called a stopping time, and the above definition requires that it is decidable whether or not T = n with a knowledge only of the past and present, Xo, X₁,..., Xn, and with no further information about the future. Show that P(XT+m=j | Xk = xk for 0 ≤ k < T, XT = i) = P(XT+m = j | XT = i) for m≥ 0, i, je S, and all sequences (xx) of states.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Strong Markov property. Let X be a Markov chain on S, and let T be a random variable taking values in {0, 1, 2,...} with the property that the indicator function I{T=n), of the event that T = n, is a function of the variables X₁, X2, ..., Xn. Such a random variable T is called a stopping time, and the above definition requires that it is decidable whether or not T = n with a knowledge only of the past and present, Xo, X1,..., Xn, and with no further information about the future. Show that P(XT+m=j | Xk = xk for 0 ≤ k < T, XT = i) = P(XT+m = j | XT = i) for m≥ 0, i, je S, and all sequences (xx) of states.
Transcribed Image Text:Strong Markov property. Let X be a Markov chain on S, and let T be a random variable taking values in {0, 1, 2,...} with the property that the indicator function I{T=n), of the event that T = n, is a function of the variables X₁, X2, ..., Xn. Such a random variable T is called a stopping time, and the above definition requires that it is decidable whether or not T = n with a knowledge only of the past and present, Xo, X1,..., Xn, and with no further information about the future. Show that P(XT+m=j | Xk = xk for 0 ≤ k < T, XT = i) = P(XT+m = j | XT = i) for m≥ 0, i, je S, and all sequences (xx) of states.
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