The name refers to the fact that 0 1 2 0 is a triangle

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Unformatted text preview: = {2, 4, . . .}, so the greatest common divisor is 2. Similarly, in Example 1.22, {n 1 : pn (x, x) > 0} = {5, 10, . . .}, so the greatest common divisor is 5. As the next example shows, things aren’t always so simple. Example 4.4. Triangle and square. Consider the transition matrix: 2 1 0 1 2 3 2 0 1 0 0 0 0 1 0 0 0.5 0 0 0 0 1 0 0 0 0 1 1 0 0 0.5 0 0 0 2 0 0 0 1 0 0 3 0 0 0 0 1 0 In words, from 0 we are equally likely to go to 1 or 1. From 1 we go with probability one to 2 and then back to 0, from 1 we go to 2 then to 3 and back to 0. The name refers to the fact that 0 ! 1 ! 2 ! 0 is a triangle and 0 ! 1 ! 2 ! 3 ! 0 is a square. 24 CHAPTER 1. MARKOV CHAINS -1 • 1/2 ? • -2 0 • ✓6 • 3 1/2 1 -• ? • 2 Clearly, p3 (0, 0) > 0 and p4 (0, 0) > 0 so 3, 4 2 I0 . To compute I0 the following is useful: Lemma 1.15. Ix is closed under addition. That is, if i, j 2 Ix , then i + j 2 Ix . Proof. If i, j 2 Ix then pi (x, x) > 0 and pj (x, x) > 0 so pi+j (x, x) pi (x, x)pj (x, x) > 0...
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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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