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09831HW5solutions

09831HW5solutions - HOMEWORK 5 SOLUTIONS Problem 1 a We can...

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HOMEWORK 5 SOLUTIONS Problem 1 a) We can show this immediately using Corollary 6.4 from Chapter 1 of Durret. b)Let X = c and for any > 0 define the open interval A = ( c - , c + ). The boundary ∂A = { c - , c + } , therefore P { X ∂A } = 0 . By the Portmanteau Theorem 1 = P { X A } = lim n →∞ P { X n A } = lim n →∞ P {| X n - c | < } Problem 2: Observe that f n is always nonnegative. To show that is a density function we need to compute the integral: Z 1 0 1 - cos(2 πnx ) dx = x - sin(2 πnx ) 2 πn 1 x =0 = 1 . For x [0 , 1] , define F n ( x ) = x - sin(2 πnx ) 2 πn . Observe that for all x [0 , 1], (1) lim n →∞ F n ( x ) = F ( x ) = x which is the distribution function of a uniform [0,1] r.v. We actually proved that for any point of continuity of F ( x ) the convergence (1) holds, therefore the μ n converge weakly to the Lebesque measure. Problem 3: Let σ 2 = V ar ( X 1 ) . Classical CLT implies that S n σ n = X N (0 , 1) . Let M > 0 . The set ( σM, + ) is open and so by the Portmanteau Theorem, (2) lim inf n →∞ P S n σ n > M P { X > M } = 1 - Φ( M ) > 0 .
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