hw 9 - n-1 log P ( n-1 S n a ) , a > 1 and n-1...

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205B homework, week 8; due Thursday March 19 1. Prove the following slight extension of Azuma’s inequality. Let ( M n ) be a martingales such that | M n - M n - 1 | ≤ K n for constants K n . Then for x > 0 P ( | M n - M 0 | ≥ x ) 2 exp - 1 2 x 2 / n X i =1 K 2 i ! . 2. Suppose you have n items, the i ’th item having size V i > 0 and reward 0 W i B for constant B , where the r.v.’s ( V i ,W i ) are all independent. You have a box of fixed size c . You pack items into the box (subject to the constaint that the sum of sizes of packed objects is at most c ), choosing the subset of items which maximizes the total reward (that is i W i , summed over the packed items). Write X for this maximum total reward. Show P ( | X - EX | ≥ x ) 2 exp( - x 2 / (2 nB 2 )) , x > 0 . 3. Let S n = n i =1 X i , where ( X i ) are i.i.d. with exponential(1) distribution. Use the large deviation theorem to get explicit limits for
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Unformatted text preview: n-1 log P ( n-1 S n a ) , a > 1 and n-1 log P ( n-1 S n a ) , a < 1. 4. Oriented rst passage percolation. Consider the lattice quadrant { ( i,j ) : i,j } with directed edges ( i,j ) ( i +1 ,j ) and ( i,j ) ( i,j +1). Associate to each edge e an exponential(1) r.v. X e , independent for dierent edges. For each directed path of length d started at (0 , 0), let S = edges e in path X e . Let H d be the minimum of S over all such paths of length d . It can be shown that d-1 H d c a.s., for some constant c . Give explicit upper and lower bounds on c . [Hint: use result of previous question for lower bound.] 9...
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