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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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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