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 May 13th, 2013, 09:57 AM #1 Newbie   Joined: May 2013 Posts: 3 Thanks: 0 A serious Random walk needs rigorous proof! PLEASE HELP! The random walk is the stochastic process Sn := sum_{i=1}^{n} e_i with real-valued, independent, and identically distributed e_i. We say that the random walk is symmetric if the law of e_k is same as that of -e_k. For a symmetric random walk, show the following (1) P(Sn - Sk >= 0) >= 1/2 for k = 1; ... ; n. (2) Fix x > 0. Let  t= inf k >=1 : Sk > x and show that P(Sn > x) >= sum_{i=1}^{n} P(t=k, Sn-Sk>=0) >= 1/2 * sum_{i=1}^{n} P(t=k) (3) Deduce that for any x > 0 and any n P(max_{k=1}^n Sk>x) <=2* P(Sn>x)

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