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# (15 points) Let Zi,i Z 1, beasequence of i. random variables, P (Z,- = 1) = P(Z,- = 1) = 1/2 fori 3 1.We set n So=0,Sn = ZZleZ 1, k=1 that is S is...

THIS IS BASIC PROBABILITY AND MARTINGALE PROBLEM.

1. (15 points) Let Zi,i Z 1, beasequence of i.i.d. random variables, P (Z,- = 1) =
P(Z,- = —1) = 1/2 fori 3 1.We set n
So=0,Sn = ZZleZ 1,
k=1 that is S is symmetric random walk on Z. Let a be a strictly positive integer and
t=inf{n 30zSn =a}, the ﬁrst a—visit time. Let .7” = (21,...,Zn), n 2 1.
a) Show that for r E R, erSn X’ = —, &gt; 0,
&quot; (coshr)&quot; n — is a In martingale. Denote n /\ r = min {11, 1'}. Show that X &quot; n 3 0, is a bounded HA1.”
martingale.

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