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Lec3 - Stat 150 Stochastic Processes Spring 2009 Lecture 3...

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Stat 150 Stochastic Processes Spring 2009 Lecture 3: Martingales and hitting probabilities for random walk Lecturer: Jim Pitman A sequence of random variables S n ,n ∈ { 0 , 1 , 2 ,... } , is a martingale if 1) E | S n | < for each n = 0 , 1 , 2 ,... 2) E ( S n +1 | ( S 0 ,S 1 ,...,S n )) = S n E.g., S n = S 0 + X 1 + ··· + X n , where X i are differences, X i = S i - S i - 1 . Observe that ( S n ) is a martingale if and only E ( X n +1 | S 0 ,S 1 ,...,S n ) = 0. Then say ( X i ) is a martingale difference sequence. Easy example: let X i be independent random variables (of each other, and of S 0 ), where E ( X i ) 0 = X i are martingale differences = S n is a martingale. Some general observations : If S n is a martingale, then E ( S n ) E ( S 0 ) (Constant in n ). Proof : By induction, suppose true for n , then E ( S n +1 ) = E ( E ( S n +1 | ( S 0 ,S 1 ,...,S n ))) = E ( S n ) = E ( S 0 ) Illustration : Fair coin tossing walk with absorption at barriers 0 and b . Process:
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Lec3 - Stat 150 Stochastic Processes Spring 2009 Lecture 3...

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