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p(x, y )g (y )
y If Px (VA < 1) > 0 for all x 2 C , then g (x) = Ex (VA ). Since g (x) = 0 for x 2 A the equation for g can be written for x 2 C as
g (x) = 1 + X r(x, y )g (y ) y so if we let ~ be a column vector consisting of all 1’s then the last equation says
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(I r)g = ~ and
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g = (I r) 1~ .
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Since (I r) 1 = I + r + r2 + . . ., for x, y 62 A, (I
number of visits to y starting from x. 1.12 r) 1 (x, y ) is the expected Exercises
Understanding the deﬁnitions 1.1. A fair coin is tossed repeatedly with results Y0 , Y1 , Y2 , . . . that are 0 or 1
with probability 1/2 each. For n 1 let Xn = Yn + Yn 1 be the number of 1’s
in the (n 1)th and nth tosses. Is Xn a Markov chain? 63 1.12. EXERCISES 1.2. Five white balls and ﬁve black balls are distributed in two urns in such a
way that each urn contains ﬁve balls. At each step we draw one ball from each
urn and exchange them. Let Xn be the number of white balls in the left urn at
time n. Compute the transition probability for Xn .
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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