# Chapter 10 - Chapter 10 Random Walks and Other Applications...

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1 Chapter 10 Random Walks and Other Applications Teacher: £ ª Office: 805 Tel: ext. 31822 Email: [email protected]

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2 10-1 Random Walks 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 0 0 = 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 life b 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 + )

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4 Let n r = S represent the event “at stage n q 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 number 1 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 + - = + (10-4) h binomial distribution 1 0 1 2 n r + integer between 0 and n q n q r even or odd
5 Return to the origin 0 n = S S i.e. 0 r = in Eq.(10-4) 1 2 n k = in Eq.(10-2) h n q even j 2n th trial 1 { } 2 2 2 0 ( ) n n n n P pq u n = = S S S event 1 1

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6 The Wiener Process 1 1 2 ( ) n n nT = = + + + X S X X X random walk 1 take step size s ± { } 0 i E = X S { } 2 2 i E s = X { } ( ) 0 E nT = X S { } 2 2 ( ) E nT ns = X ( 1 independence of i X )
7 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 np q 1 2 p q = = 2 m k n = - (10-2) 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 =

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8 Wiener Process 1 n → ∞ 0 T 2 s T α = define ( ) lim ( ) t t = W X S 0 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 0 t form 1 RV 1 0 t = - Z t S 1 t S 0 t q point 1 Fig. 10-8 1 1 ( ) z f z e λ λ - = Z S ( ) 1 z F z e λ - = - Z S 0 z

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10 ° { } ( ) F z P z = Z Z { } 0 0 ( , ) 0 P t t z = + n (i.e., 0 ( ) t z + ) { } 0 0 1 ( , ) 0 P t t z = - + = n 0 ( ) 1 0! z e z λ λ - = - 1 z e λ - = - ( ) z f z e λ λ - = Z
11 ° 0 1 t - = - w t S 1 - t S 0 t point1 ( ) w f w e λ λ - = W S ( ) 1 w F w e λ - = - W S 0 w h 0 n n t = - X t S n t S 0 t q n th point1 1 ( ) ( 1)! n n x n f x x e n λ λ - - = - Gamma distribution

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12 Proof 1 { } { } 0 0 ( ) 1 ( , ) n n F x P x P t t x n = = - + < X n 1 0 ( ) 1 ! k n x k x e k λ λ - - = = - 1 1 n q 0 ( , ) t x q q ( ) n F x q x q 1 1 1 0 1 ( ) ( ) ( ) ( ) ! ! k k n n x n k k x k x f x e k k λ λ λ λ - - - - = = = - 1 2 0 0 ( ) ( ) ( ) ! ! k k n n x k k x x e k k λ λ λ λ - - - = = = - 1 ( 1)! n n x x e n λ λ - - = -
13 h 1 1 n n n n - - = - = - X X X t t ( 1 1 points 1 ) 1 1 ( ) x X f x e λ λ - = Poisson Points L 4 models 1 Photon count 1 electron emission 1 telephone calls 1 data communication 1 visits to a doctor 1 arrivals at a park 1 L4 general form 1 Poisson Points 1 model 1 1

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14 Given a sequence n w S i.i.d. RVs with density ( ) w f w e λ λ - = 0 w form a set of points n t S as 1 2 n n = + + + t w w w with 0 0 = t S origin 1 1 n t S Poisson distributed with parameter λ h N points in an interval of length T q 1 N q q q Poisson points with parameter / N T q N q T → ∞ exactly
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