To introduce this tool we need one more notion we say

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Unformatted text preview: j E [(Mn X Mk )Mj |Av ] E [(Mn Mk )|Av ] = 0 x proves the desired formula. The last result has the following useful corollary E (Mn M0 )2 = n X E (Mk Mk 2 1) (5.6) k=1 Proof. Expanding out the square of the sum E n X k=1 Mk Mk 1 ! 2 = n X k=1 +2 E (Mk X Mk E [(Mk 1j<kn and the second sum vanishes by Lemma 5.8. 2 1) Mk 1 )(Mj Mj 1 )] 165 5.3. GAMBLING STRATEGIES, STOPPING TIMES 5.3 Gambling Strategies, Stopping Times The first result should be intuitive if we think of supermartingale as betting on an unfavorable game: the expected value of our fortune will decline over time. Theorem 5.9. If Mm is a supermartingale and m n then EMm E Mn . Proof. It is enough to show that the expected value decreases with each time step, i.e., EMk E Mk+1 . To do this, we will again use the notation from (5.5) Av = {Xn = xn , Xn 1 = xn 1 , . . . , X0 = x0 , M0 = m} and note that linearity in the conditioning set (Lemma 5.3) and the definition of conditional expectation imply X E (Mk+1 Mk ) = E (Mk+1 Mk ; Av ) v = X P (Av )E (Mk+1 Mk |Av ) 0 v since supermartingales have E (Mk+1 Mk |...
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