6.262.Lec26

# 6.262.Lec26 - DISCRETE STOCHASTIC PROCESSES Lecture 26...

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Lecture 26 - 5/12/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 26 Martingales: Sections 7.6 - 7.10 Review: Martingales and Submartingales Urn Example Jensen’s Inequality Stopping Times and Stopped Submartingales and Supermartingales Kolmogorov Submartingale Inequality Applications of Kolmogorov Submartingale Inequality: Urn problem Random walk bound Branching processes Corollary 1: Kolmogorov Martingale Inequality Corollary 2: Kolmogorov’s Random Walk Inequality Strong Law of Large Numbers & Proof Martingale Convergence Theorem Application to Branching Processes

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Lecture 26 - 5/12/2010 Discrete Stochastic Processes 2 M ARTINGALES Definition : A martingale Z n ; n 1 { } is a stochastic process with the properties that EZ n [] < ∞ for all n and n Z n 1 = z n 1 , Z n 2 = z n 2 , K , Z 1 = z 1 [ ] = z n 1 for all n > 1 and all z 1 , z 2 , K , z n 1 . Definition: A stochastic process Z n ; n 1 { } is a submartingale if n <∞ for all n > 1 and if 1 1 , , 2 , 1 n Z Z n Z n Z n Z E K .
Lecture 26 - 5/12/2010 Discrete Stochastic Processes 3 Nice (Bounded) Example We begin at n=0 with an urn with 1 red ball and 1 black ball. At each time we pick a ball at random, return it and another ball of the same color to the urn, stir up the balls in the urn, and continue. Then f n , the fraction of red balls at time n, is a martingale. Why? To see this, let be the number of red balls in the urn at time n. and 1 r nn 1 22 n r1 r [ | ]( 1 ) ( ) (n+2) ( 3) (n+2) ( () ( ( 2 ) ) r . ( 2)( (n+2) n n n rr Ef f r n r f + + = +− = ++ + == n r Then (n+2) n f =

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Lecture 26 - 5/12/2010 Discrete Stochastic Processes 4 Jensen's Inequality : If h x ( ) is convex and if X is a rv with [ ] < X E , then h(E[X]) E [h(X)]. Why: Let x 1 = E[X] and choose c so that h(x 1 ) + c (x - x 1 ) h(x) for all x. Then h(E[X]) + c (X( ω ) - E[X]) h(X( ω )), all ω ε , so E [ h(E[X]) + c (E[X] - E[X]) ] E [h(X)], i.e., h(E[X]) E [h(X)]. Mnemonic: If h(x) = x 2 , then h(E[X]) = (E[X]) 2 (E[X]) 2 + δ X 2 = E[X 2 ] = E[h(X)]
Lecture 26 - 5/12/2010 Discrete Stochastic Processes 5 Theorem : If {Z n , n 1} is a martingale and h is convex, then {h(Z n) , n 1} is a submartingale. Examples : If {Z n , n 1} is a martingale, then {|Z n| , n 1} is a submartingale, {(Z n ) 2 n 1} is a submartingale, and {e rZn , n 1} is a submartingale. Corollary: If {Z n , n 1} is a martingale, then Var (Z n ), n 1, is nondecreasing.

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Lecture 26 - 5/12/2010 Discrete Stochastic Processes 6 Stopping Times and Stopped Processes J is a stopping time for the random process {Z n , n 1} if the indicator function I J=n is a function of (at most) Z 1 , Z 2 ,±---Z n , and J is a (nondefective) random variable. If J is a defective random variable, then J is a defective stopping time for the process {Z n , n 1}. The stopped process {Z n *, n 1} becomes constant at n = J and afterwards, i.e., Z n * = Z n , 1 n J, and Z n * = Z J , J n Lemma: Let J be a possibly defective stopping rule for the random process {Z n , n 1}. Then {Z n , n 1} is a martingale {Z* n , n 1} is a martingale {Z n , n 1} is a submartingale {Z* n , n 1} is a submartingale
Lecture 26 - 5/12/2010 Discrete Stochastic Processes 7 Properties Martingale

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6.262.Lec26 - DISCRETE STOCHASTIC PROCESSES Lecture 26...

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