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lecture3

# lecture3 - STAT 150 CLASS NOTES Onur Kaya 16292609...

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STAT 150 CLASS NOTES Onur Kaya 16292609 May 18, 2006 Martingales : A sequence of random variables ( M n ) is a martingale relative to the sequence ( X n ) if: 1. M n is some measurable function of X 1 , X 2 , ..... , X n 2. E [ M n +1 | X 1 , X 2 , ..... , X n ] = M n Notice that (1) + (2) E [ M n +1 | M 1 , M 2 , ..... , M n ] = M n . Variations: If we replace ”=” in (2) by ” then it is a submartingale a favorable game ” then it is a supermartingale unfavorable game Fundamental: I. If M n is a MG then E ( M n ) = E ( M 0 ) is constant SubMG then E ( M n ) is increasing SuperMG then E ( M n ) is decreasing II. Take a MG M n and T a stopping time relative to the sequence X n . Then, T = n is a function of X 1 , X 2 .... X n . Look at the process ˆ M n := M n if n T M T if n > T ˆ M n := M n V T Then, ˆ M n is a MG relative to X n . Application: Back to Gambler’s Ruin Setup. X i = 1 w.p. 1/2. 1

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2 M 0 = a , M n = a + X 1 + X 2 + .... + X n T=first n s.t. M n 0 , b E [ M n V T ] = E [ M 0 ] = a E [ M n V T ] = bP ( T n, M T = b ) + 0 .P ( T n, M T = 0) + E [ M n 1( T > n )] Let n → ∞ , then the last term is 0 and P ( M T = b
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