Central Limit Theorem, Random Walk, Brownian Motion
Central Limit Theorem, Random Walk, Brownian Motion
Central Limit Theorem, Random Walk, Brownian Motion
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Computing the MGFCLT and<strong>Random</strong><strong>Walk</strong>JeffreySchenkerA GameWorking onthe ProblemInterlude:Gambler’sRuinCLTRand. <strong>Walk</strong><strong>Brownian</strong><strong>Motion</strong>Φ n (s) = Av( ) (e sW n= Av= Ave s ∑n i=1 w iAv (e sw i ) =e s +e −s2=: cosh s.Φ n (s) =)( n∏e sw ii=1)cosh s = 1 + 1 2 s2 + O(s 4 ) for s small.n∏i=1Av (e sw i ) = (cosh s)n=n∏i=1Av (e sw i)
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