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   Let {Xi} be i.i.d. random variables with P(Xi=−1) = P(Xi= 1) = 1/2. Let Sn=...

  

Let {Xi} be i.i.d. random variables with P(Xi=−1) = P(Xi= 1) = 1/2. Let Sn= 1 +X1+. . .+Xn be symmetric simple random walk with initial point S0 = 1. Find the probability that Sn eventually hits the point 0.

Hint: Define the events A={Sn= 0 for some n} and for M >1, AM = {Sn hits 0 before hitting M}.

Show that AM ↗ A.

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