One possible strategy is to choose x and y so that

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Unformatted text preview: nly if (b) ! 1 as b ! 1, giving another solution of Exercise 9.46 from Chapter 1. 5.8. Let Sn = X1 + · · · + Xn where the Xi are independent with EXi = 0 and 2 var (Xi ) = 2 . (a) Show that Sn n 2 is a martingale. (b) Let ⌧ = min{n : a2 / 2 . For simple random |Sn | > a}. Use Theorem 5.13 to show that E ⌧ 2 walk = 1 and we have equality. 5.9. Wald’s second equation. Let Sn = X1 + · · · + Xn where the Xi are independent with EXi = 0 and var (Xi ) = 2 . Use the martingale from the previous 2 problem to show that if T is a stopping time with ET < 1 then EST = 2 ET . 5.10. Mean time to gambler’s ruin. Let Sn = S0 + X1 + · · · + Xn where X1 , X2 , . . . are independent with P (Xi = 1) = p < 1/2 and P (Xi = 1) = 1 p. Let V0 = min{n 0 : Sn = 0}. Use Wald’s equation to conclude that if x > 0 then Ex V0 = x/(1 2p). 5.11. Variance of the time of gambler’s ruin. Let ⇠1 , ⇠2 , . . . be independent with P (⇠i = 1) = p and P (⇠i = 1) = q = 1 p where p < 1/2. Let Sn = S0 + ⇠1 + · · · + ⇠n . In Example 4...
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