# Consider now a random walk sn s0 x1 xn where x1

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online