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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<strong>Central</strong> <strong>Limit</strong> <strong>Theorem</strong>CLT and<strong>Random</strong><strong>Walk</strong>JeffreySchenkerA GameWorking onthe ProblemInterlude:Gambler’sRuinCLTRand. <strong>Walk</strong><strong>Brownian</strong><strong>Motion</strong>If X is a standard normal random variable thenAv(e sX ) = 1 √2π∫ ∞−∞e sX e − 1 2 X 2 dX = e 1 2 s2 .Wait! That is familiar... we had:( ) ( )lim Av e s √ 1nW n 1= lim Φ n √n s = e 1 2 s2 .n→∞ n→∞<strong>Central</strong> <strong>Limit</strong> <strong>Theorem</strong>For large n,1 √n W n is almost a standard normal randomvariable.
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