xn gives x p xt 1 z xt y t n p xn1 z

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Unformatted text preview: according to the values of X0 , . . . , Xn gives X P (XT +1 = z, XT = y, T = n) = P (Xn+1 = z, Xn = xn , . . . , X0 = x0 ) = X x2Vn x2Vn P (Xn+1 = z |Xn = xn , . . . , X0 = x0 )P (Xn = xn , . . . , X0 = x0 ) where in the second step we have used the multiplication rule P (A \ B ) = P (B |A)P (A) 12 CHAPTER 1. MARKOV CHAINS For any (x0 , . . . , xn ) 2 Vn we have T = n and XT = y so xn = y . Using the Markov property, (1.1), and recalling the definition of Vn shows the above X P (XT +1 = z, T = n, XT = y ) = p(y, z ) P (Xn = xn , . . . , X0 = x0 ) x2Vn = p(y, z )P (T = n, XT = y ) Dividing both sides by P (T = n, XT = y ) gives the desired result. 1 Let Ty = Ty and for k 2 let k k Ty = min{n > Ty 1 : Xn = y } (1.3) be the time of the k th return to y . The strong Markov property implies that the conditional probability we will return one more time given that we have returned k 1 times is ⇢yy . This and induction implies that k Py (Ty < 1) = ⇢k yy (1.4) At this point, there are two possibilities: (i) ⇢yy < 1: The...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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