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Unformatted text preview: . E.g., in Example 5.4 Mn = Xn n.
To explain the reason for our interest in martingales, we will now give a
number of examples. In what follows we will often be forced to write the
conditioning event, so we introduce the short hand
Av = {Xn = xn , Xn 1 = xn 1 , . . . , X0 where v is short for the vector (xn , . . . , x0 , m0 ) = x0 , M0 = m0 } (5.5) 162 CHAPTER 5. MARTINGALES Example 5.2. Random walks. Let X1 , X2 , . . . be i.i.d. with EXi = µ. Let
Sn = S0 + X1 + · · · + Xn be a random walk. Mn = Sn nµ is a martingale
with respect to Xn .
Proof. To check this, note that Mn+1 Mn = Xn+1 µ is independent of
Xn , . . . , X0 , M0 , so the conditional mean of the di↵erence is just the mean:
E (Mn+1 Mn Av ) = EXn+1 µ=0 In most cases, casino games are not fair but biased against the player. We
say that Mn is a supermartingale with respect to Xn if a gambler’s expected
winnings on one play are negative:
E (Mn+1 Mn Av ) 0 To help remember the direction of the inequality, note that there is nothing
“super” about a supermartingal...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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