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Unformatted text preview: 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 ) 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 M n := M n if n T M T if n > T M n := M n V T Then, M n is a MG relative to X n . Application: Back to Gamblers Ruin Setup. X i = 1 w.p. 1/2....
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This note was uploaded on 10/02/2009 for the course STAT 87528 taught by Professor Pitman,jim during the Spring '09 term at University of California, Berkeley.
- Spring '09