0 as n 1 the desired result follows example 511

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Unformatted text preview: setting, let Xm = 1 if the mth coin flip is heads and 1 if the mth flip is tails, and let Mn = X1 + · · · + Xn as the net profit of a gambler who bets 1 unit every time. Theorem 5.12. Suppose that Mn is a supermartingale with respect to Xn , Hn is predictable, and 0 Hn cn where cn is a constant that may depend on n. Then n X Wn = W0 + Hm (Mm Mm 1 ) is a supermartingale m=1 We need the condition Hn 0 to prevent the bettor from becoming the house by betting a negative amount of money. The upper bound Hn cn is a technical condition that is needed to have expected values make sense. In the gambling context this assumption is harmless: even if the bettor wins every time there is an upper bound to the amount of money he can have at time n. 167 5.3. GAMBLING STRATEGIES, STOPPING TIMES Proof. The change in our wealth from time n to time n + 1 is Wn = Hn+1 (Yn+1 Wn+1 Yn ) As in the proof of Theorem 5.9 let Av = {Xn = xn , Xn 1 = xn 1 , . . . , X0 = x0 , M0 = m0 }. Hn+1 is constant on the event Av , so Lemma 5.1 implies E (Hn+1 (Mn+1 Mn )|Av ) = Hn+1 E...
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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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