sol5 - Math 136 - Stochastic Processes Homework Set 5, Fall...

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Math 136 - Stochastic Processes Homework Set 5, Fall 2008, Due: October 29 See Victor Hu on questions on grading 1. Exercise 3.2.21. Consider the random variables b S k of Example 1.4.13. (a) Applying Proposition 3.2.6 verify that the corresponding characteristic functions are Φ b S k ( θ ) = [cos( θ/ k )] k . ANS: Let X i for i = 1 ...k be i.i.d. RVs with P ( X i = - 1) = P ( X i = 1) = 1 / 2. Then using Proposition 3.2.6 for the first equality we have Φ b S k ( θ ) = k Y i =1 Φ X i / k ( θ ) = { Φ X 1 / k ( θ ) } k = { E ( e iθX 1 / k ) } k = { ( e - iθ/ k + e iθ/ k ) / 2 } k = { cos( θ/ k ) } k (b) Recalling that δ - 2 log(cos δ ) → - 0 . 5 as δ 0, find the limit of Φ b S k ( θ ) as k → ∞ while θ IR is fixed. ANS: Note that Φ b S k ( θ ) = exp { k log[cos( θ/ k )] } . Taking δ = θ/ k and exploiting the continuity of the exponential function we get Φ b S k ( θ ) e - θ 2 / 2 . (c) Suppose random vectors X ( k ) and X in IR n are such that Φ X ( k ) ( θ ) Φ X ( θ ) as k → ∞ , for any fixed θ . It can be shown that then the laws of X ( k ) , as probability measures on IR n , must converge weakly in the sense of Definition 1.4.20 to the law of X . Explain how this fact allows you to verify the C.L.T. statement b S n L -→ G of Example 1.4.13. ANS: From the previous part we see that Φ b S k ( θ ) Φ G ( θ ) for all θ , where G is a standard normal random variable. Then what has been stated above implies that b S k L -→ G . 2. Exercise 3.2.22. Consider the random vectors X ( k ) = ( 1 k S k/ 2 , 1 k S k ) in IR 2 , where k = 2 , 4 , 6 ,... is even, and S k is the simple random walk of Definition 3.1.2, with P ( ξ 1 = - 1) = P ( ξ 1 = 1) = 0 . 5. (a) Verify that Φ X ( k ) ( θ ) = [cos(( θ 1 + θ 2 ) / k )] k/ 2 [cos( θ 2 / k )] k/ 2 , where θ = ( θ 1 2 ). ANS: Here Φ X ( k ) ( θ ) = E exp( θ 1 S k/ 2 / k + θ 2 S k / k ) and since S k = S k/ 2 + ˜ S k/ 2 where ˜ S k/ 2 is independent, identically distributed copy of S k/ 2 , we have E exp( θ 1 S k/ 2 / k + θ 2 S k / k ) = E exp[( θ 1 + θ 2 ) S k/ 2 / k ] E exp[ θ 2 S k/ 2 / k ] The required result now follows by noting that S k / k has the same distribution as b S k from Exercise 3.2.21, so their characteristic functions are equal. 1
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(b) Find the mean vector μ and the covariance matrix Σ of a Gaussian random vector X for which Φ X ( k ) ( θ ) converges to Φ X ( θ ) as k → ∞ . ANS: Same approach as in part (b) of the Exercise 3.2.21 gives Φ X ( k ) ( θ ) e - ( θ 1 + θ 2 ) 2 / 4 e - θ 2 2 / 4 . We now need μ and Σ such that exp[ - ( θ , Σ θ ) / 2 + i ( θ )] = exp[( - θ 2 1 / 2 - θ 1 θ 2 - θ 2 2 ) / 2] which gives μ = (0 , 0), Σ 11 = Σ 12 = Σ 21 = 1 / 2 and Σ 22 = 1. (c) Upon appropriately generalizing what you did in part (b), I claim that the
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sol5 - Math 136 - Stochastic Processes Homework Set 5, Fall...

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