The two are numbered in the same sequence to make

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Unformatted text preview: follows, lemmas are a means to prove the more important conclusions called theorems. The two are numbered in the same sequence to make results easier to find. Lemma 1.4. If x ! y and y ! z , then x ! z . Proof. Since x ! y there is an m so that pm (x, y ) > 0. Similarly there is an n so that pn (y, z ) > 0. Since pm+n (x, z ) pm (x, y )pn (y, z ) it follows that x ! z. Theorem 1.5. If ⇢xy > 0, but ⇢yx < 1, then x is transient. 14 CHAPTER 1. MARKOV CHAINS Proof. Let K = min{k : pk (x, y ) > 0} be the smallest number of steps we can take to get from x to y . Since pK (x, y ) > 0 there must be a sequence y1 , . . . yK 1 so that p(x, y1 )p(y1 , y2 ) · · · p(yK 1 , y ) > 0 Since K is minimal all the yi 6= y (or there would be a shorter path), and we have Px (Tx = 1) p(x, y1 )p(y1 , y2 ) · · · p(yK 1 , y )(1 ⇢yx ) > 0 so x is transient. We will see later that Theorem 1.5 allows us to to identify all the transient states when the state space is finite. An immediate consequence...
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