Consider now a random walk sn s0 x1 xn where x1

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Unformatted text preview: (Mn+1 Mn |Av ) 0 verifying that Wn is a supermartingale. Arguing as in the discussion after Theorem 5.9 the same result holds for submartingales and for martingales with only the assumption that |Hn | cn . Though Theorem 5.12 may be depressing for gamblers, a simple special case gives us an important computational tool. To introduce this tool, we need one more notion. We say that T is a stopping time with respect to Xn if the occurrence (or nonoccurrence) of the event {T = n} can be determined from the information known at time n, Xn , Xn 1 . . . X0 , M0 . Example 5.8. Constant betting up to a stopping time. One possible gambling strategy is to bet $1 each time until you stop playing at time T . In symbols, we let Hm = 1 if T m and 0 otherwise. To check that this is an admissible gambling strategy we note that the set on which Hm is 0 is {T m}c = {T m 1} = [m 1 {T = k } k=1 By the definition of a stopping time, the event {T = k } can be determined from the values of M0 , X0 , . . . , Xk . Since the union is over k m 1, Hm can be determined from th...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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