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Unformatted text preview: 1 Chapter 10 Random Walks and Other Applications Teacher: £ ª Office: 805 Tel: ext. 31822 Email: [email protected] 2 101 Random Walks h& independent RV sequence 1 +1 1 1 1 p 1 1 q p =  Bernoulli trials 1 2 , , X X 1 2 n n = + + + S X X X = S S S accumulated positive or negative excess at the n th trial h random walk model 1 takes a unit step or regular interval up or down 1 i X S S n S S S g / at the n th step1 1 2 p q = = symmetric1 unsymmetrical 3 ¨T £”“b& real lifeb model Motion of gas molecules Thermal noise Stock value h model b events Return to the origin The first return to the origin Waiting time for the first gain (first visit to 1 + ) 4 Let n r = S represent the event “at stage nq the particle is at the point r ”1 , n r P ” { } , k n k n r n n P P r p q k = = S S 1 k q successes 1 number1 n k failures 1 number h ( ) 2 r k n k k n = = 1 ( ) 2 k n r = + 1 ( )/ 2 ( )/ 2 , ( )/ 2 n r n r n r n P p q n r + = + (104) h binomial distribution1 01 2 n r + integer between and n q n q r even or odd 5 Return to the origin n = S S i.e. r = in Eq.(104)1 2 n k = in Eq.(102) h n q evenj ” 2n th trial 1 { } 2 2 2 ( ) n n n n P pq u n = = S S S event 1 1 6 The Wiener Process 1 1 2 ( ) n n nT = = + + + X S X X X random walk 1 take step size s ± { } i E = X S { } 2 2 i E s = X ⇒ { } ( ) E nT = X S { } 2 2 ( ) E nT ns = X ( 1 independence of i X ) 7 h& 2 ( ) /2 1 2 k n k k np npq n p q e k npq π 2245 for n q k in the npq vicinity of npq 1 2 p q = = 2 m k n = (102)1 { } 2 / 2 1 ( ) / 2 m n P nT ms e n π = X S 2 2 m n k = + 2 m k np = m of the order of n ⇒ { } ( ) ( / ) P t ms G m n ≤ X S for nT T t nT < ≤ i.e., Gaussian cdf with zero mean, n σ = 8 Wiener Process 1 n → ∞ T → 2 s T α = define ( ) lim ( ) t t = W X S T → t nT = Wiener Process 1 2 /2 1 ( , ) (0, ) 2 t f t e N t t ϖ α ϖ α πα = = 1 1st order statistics 9 10.2 Poisson Points and Shot noise h Poisson Points i t ” t form 1 RV 1 t = Z t S 1 t S t ” q point1 Fig. 108 1 1 ( ) z f z e λ λ = Z S ( ) 1 z F z e λ = Z S z 10 h& & { } ( ) F z P z = ≤ Z Z { } ( , ) P t t z = + n (i.e., ” ( ) t z + ) { } 1 ( , ) P t t z =  + = n ( ) 1 0! z e z λ λ =  1 z e λ =  ( ) z f z e λ λ = Z 11 h& & 1 t = w t S 1 t S t point1 ( ) w f w e λ λ = W S ( ) 1 w F w e λ = W S w h n n t = X t S n t S t q n th point1 1 ( ) ( 1)! n n x n f x x e n λ λ = Gamma distribution 12 Proof1 { } { } ( ) 1 ( , ) n n F x P x P t t x n = ≤ =  + < X n 1 ( ) 1 !...
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

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