SRW - is comprised of dependent random variables. (d)...

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Statistics 351 (Fall 2007) Simple Random Walk Suppose that Y 1 , Y 2 , . . . are i.i.d. random variables with P { Y 1 = 1 } = P { Y 1 = - 1 } = 1 / 2, and define the discrete time stochastic process { S n , n = 0 , 1 , . . . } by setting S 0 = 0 and S n = n X i =1 Y i . We call { S n , n = 0 , 1 , . . . } a simple random walk (SRW). One useful way to visualize a SRW is to graph its trajectory n 7→ S n . In other words, plot the pairs of points ( n, S n ), n = 0 , 1 , 2 , . . . and join the dots with straight line segments. (a) Suppose that a realization of the sequence Y 1 , Y 2 , . . . produced - 1 , - 1 , - 1 , 1 , 1 , - 1 , 1 , 1 , - 1 , 1 , 1 , 1 , - 1 , 1 , 1 . Sketch a graph of the resulting trajectory n 7→ S n of the SRW. The theory of martingales is extremely useful for analyzing stochastic processes. Recall that a stochastic process { X n , n = 0 , 1 , 2 , . . . } is called a martingale if E ( X n +1 | X n ) = X n for all n . (b) Review the proof that both S n and S 2 n - n are martingales. (c) Use the fact that E ( X n ) = E ( X 0 ) for any martingale to compute E ( S 2 n ). Even though the underlying sequence Y 1 , Y 2 , . . . is comprised of independent random vari- ables, the sequence S 1 , S 2 , S 3 , . . .
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Unformatted text preview: is comprised of dependent random variables. (d) Compute cov( S n , S n +1 ), n = 1 , 2 , . . . . The transition probabilities describe the probability that the process is at a given position at a given time. One quantity of interest is the probability that the SRW is back at the origin at time n ; that is, P { S n = 0 } . Notice that the SRW can only be at the origin after an even number of steps since the only way for it to be at 0 is for there to have been an equal number of steps to the left as to the right. This means it is notationally easier to work with P { S 2 n = 0 } , n = 0 , 1 , 2 , . . . . (e) Determine an expression for P { S 2 n = 0 } , n = 0 , 1 , 2 , . . . . (f) Try and determine an expression for P { S 2 n = x } where | x | 2 n has even parity. (That is, x is an even integer between-2 n and 2 n .)...
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This note was uploaded on 03/26/2012 for the course STAT 351 taught by Professor Michaelkozdron during the Fall '08 term at University of Regina.

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