The strong law of large numbers implies that on va 1

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Unformatted text preview: ll apply the results from the previous section to rederive some of the results from Chapter 1 about hitting probabilities and exit times for random walks. To motivate the developments we begin with a simple example. Let X1 , X2 , . . . Xn be independent with P (Xi = 1) = P (Xi = 1) = 1/2, let Sn = S0 + X1 + · · · + Xn , and let ⌧ = min{n : Sn 62 (a, b)}. To quickly derive the exist distribution it is tempting to argue that since Sn is a martingale and ⌧ is a stopping time x = Ex S⌧ = aPx (S⌧ = a) + b(1 P (S⌧ = a)) and then solve to conclude Px (S⌧ = a) = b b x a (5.10) This formula is correct, but as the next example shows, we have to be careful. Example 5.9. Bad Martingale. Suppose x = 1, let Va = min{n : Sn = 0} and T = V0 . We know that P1 (T < 1) but E1 ST = 0 6= 1 The trouble is that P1 (VN < V0 ) = 1/N so the random walk can visit some very large values before returning to 0. The fix for this problem is the same in all the examples we consider. We have a martingale Mn and a stopping time T . We use...
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