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hw8sol

# hw8sol - Math 280A Fall 2011 Homework 8 Solutions 1 In this...

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Unformatted text preview: Math 280A, Fall 2011 Homework 8 Solutions 1. In this problem, the random variables X 1 , X 2 , . . . and X are defined on a probability space (Ω , F , P ) with the special property that Ω has only finitely many points, say Ω = { ω 1 , ω 2 , . . . , ω N } , and that each singleton { ω k } is an element of F with P [ { ω k } ] > for each k . Show that X n P → X if and only if X n → X a.s. Solution. Since almost sure convergence implies convergence in probability quite generally, we need only concern ourselves with the reverse implication. Thus assume that X n P → X . From class discussion (Dominated Convergence) we know that this means that (1 . 1) lim n E [min( | X n − X | , 1)] = 0 . But, defining δ := min( P [ { ω 1 } ] , . . . , P [ { ω N } ]) > 0, we have (1 . 2) E [min( | X n − X | , 1)] = N summationdisplay k =1 min( | X n ( ω k ) − X ( ω k ) | , 1) P [ { ω k } ] ≥ δ N summationdisplay k =1 min( | X n ( ω k ) − X ( ω k ) | , 1) Therefore, (1.1) implies that each term in the sum on the right side of (1.2) converges to 0 as n → ∞ . That is, lim n min( | X n ( ω k ) − X ( ω k ) | , 1) = 0 , ∀ k = 1 , 2 , . . ., N. This in turn implies that lim n | X n ( ω k ) − X ( ω k ) | = 0 , ∀ k = 1 , 2 , . . ., N, which means that X n ( ω ) → X ( ω ) for every ω ∈ Ω. 2. Suppose that X n P → X and Y n P → Y as n → ∞ . (a) Let f : R → R be a continuous function. Show that f ( X n ) P → f ( X ) . (b) Show that X n + Y n P → X + Y and that X n Y n P → XY . Solution. (a) Suppose X n P → X and that f : R → R is continuous. Most of the difficulty here stems from the fact that f need not be uniformly continuous. Fix ǫ > 0 and η > 0. We will produce n = n ( ǫ, η ) such that n ≥ n = ⇒ P [ | f ( X n ) − f ( X ) | > ǫ ] < η. First choose N ∈ N so large that P [ | X | ≤ N − 1] ≥ 1 − η/ 2. The restriction of f to the compact interval [ − N, N ] is then uniformly continuous, so there exists δ > 0 such that (2 . 1) | x | ≤ N, | y | ≤...
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hw8sol - Math 280A Fall 2011 Homework 8 Solutions 1 In this...

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