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May 13th, 2013, 10:57 AM   #1
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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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