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Unformatted text preview: Chap. 10 Random Walks and other applications 10.1 Random Walks Let the values of an independent RV sequence be +1 and 1 with probabilities p for +1 and 1 q p =  for 1. An example is the Bernoulli trials 1 2 , , X X L . Define 1 2 n n = + + + S X X X L with = S . It is the accumulated positive or negative excess at the n th trial. A random walk model takes a unit step or regular interval up or down determined by i X . n S represents the location it stays at the n th step. When 1 2 p q = = , it is symmetric; otherwise it is unsymmetrical. Many physical phenomena and real life activities conform to the model. For instances: Motion of gas molecules Thermal noise Stock value The model can be used to study many events like Return to the origin The first return to the origin Waiting time for the first gain (first visit of 1 + ) Let n r = S represent the event of “at time n , the particle is at the point r ”. The probability of the event is { } , k n k n r n n P P r p q k = = S @ , where k is the number of successes and n k is the number of failures. Let ( ) 2 r k n k k n = = and 1 ( ) 2 k n r = + . Then ( )/2 ( )/ 2 , ( ) / 2 n r n r n r n P p q n r + = + (104) It is a binomial distribution. Its value is 0 except when 2 n r + is an integer between 0 and n . So n and r must be even or odd simultaneously. The event “Return to the origin”: The event can be specified by n = S , i.e. r = in Eq.(104), or 2 n k = in Eq.(10 2). Since n must be even, we perform “change variable” to discuss the event at the 2nth trial. Its probability can be expressed by { } 2 2 2 ( ) n n n n P pq u n = = S @ . The Wiener Process Let 1 2 ( ) n n nT = = + + + X S X X X L be a random walk which takes step size s q . Since { } i E = X and { } 2 2 i E s = X , we have { } ( ) E nT = X and { } 2 2 ( ) E nT ns = X (resulted from the independence of i X ) From 2 ( ) / 2 1 2 k n k k np npq n p q e k npq π 2245 for very large n and k in the npq vicinity of np , when 1 2 p q = = and 2 m k n = , we have { } 2 / 2 1 ( ) / 2 m n P nT ms e n π = X ; . (Note that a constant must be inserted when we perform changing variable from k to m ). Here 2 2 m n k = + , 2 m k np = , and m is in the order of n . This implies { } ( ) ( / ) P t ms G m n q X ; i.e. Gaussiam with zero mean and n σ = for nT T t nT < . Wiener Process When n q and T q , define ( ) lim ( ) t t = W X with t = nT . The process is referred to as the Wiener Process. Its 1st order statistics is 2 / 2 1 ( , ) (0, ) 2 w t f w t e N t t α α πα = = , 2 s t α = 10.2 Poisson Points and Shot noise (We skip this section) By using Poisson Points i t and a fixed time point t , we can form a RV 1 = Z t t , where 1 t is the first point appear at the right of t . Fig. 108 shows the concept of the RV. Its pdf and cdf can be expressed by ( ) z f z e λ λ = Z and ( ) 1 z F z e λ =  Z for z The derivations are given below: {...
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 Fall '08
 SinHorngChen
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