Unformatted text preview: n
m=n Using the assumptions ET < 1 and E X  < 1 one can prove that the righthand side tends to 0 and complete the proof. However the details are somewhat
complicated and are not enlightening so they are omitted. Our next two examples are applications of the exponential martingale in
Example 5.6:
Example 5.12. Leftcontinuous random walk. Suppose that X1 , X2 , . . .
are independent integervalued random variables with EXi > 0, P (Xi
1) =
1, and P (Xi = 1) > 0. These walks are called leftcontinuous since they
cannot jump over any integers when they are decreasing, which is going to the
left as the number line is usually drawn. Let (✓) = exp(✓Xi ) and deﬁne ↵ < 0
by the requirement that (↵) = 1. To see that such an ↵ exists, note that (i)
(0) = 1 and
0 (✓) = d
Ee✓xi = E (xi e✓xi )
d✓ so 0 (0) = Exi > 0 and it follows that (✓) < 1 for small negative ✓. (ii) If ✓ < 0, then (✓)
e ✓ P (xi = 1) ! 1 as ✓ ! 1. Our choice of ↵ makes exp(↵Sn ) a martingale....
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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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