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May 13th, 2013, 04:24 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 realvalued, 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, SnSk>=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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