Multiplying by 1 we see theorem 510 if mm is a

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Unformatted text preview: e. The definition traces its roots to the notion of superharmonic functions whose values at a point exceed the average value on balls centered around the point. If we reverse the sign and suppose E (Mn+1 Mn |Av ) 0 then Mn is called a submartingale with respect to Xn . A simple modification of the proof for Example 5.2 shows that if µ 0, then Sn defines a supermartingale, while if µ 0, then Sn is a submartingale. The next result will lead to a number of examples. Theorem 5.5. Let Xn be a Markov chain with transition probability p and let f (x, n) be a function of the state x and the time n so that f (x, n) = X p(x, y )f (y, n + 1) y Then Mn = f (Xn , n) is a martingale with respect to Xn . In particular if P h(x) = y p(x, y )h(y ) then h(Xn ) is a martingale. Proof. By the Markov property and our assumption on f E (f (Xn+1 , n + 1)|Av ) = X p(xn , y )f (y, n + 1) = f (xn , n) y which proves the desired result. The next two examples begin to explain our interest in Theorem 5.5. Example 5.3. Gambler’s ruin. Let X1 , X2 , . . . be independent with P (Xi =...
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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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