hw7sol - Stat 310A/Math 230A Theory of Probability Homework...

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Stat 310A/Math 230A Theory of Probability Homework 7 Solutions Andrea Montanari Due on November 18, 2010 Exercises on weak convergence of distributions Exercise [3.2.8] (a). It is easy to see that Y n D -→ c if and only if Y n p c (for example, this is a consequence of Theorem 3.2.10). Assuming the latter, we proceed to show that X n + Y n D -→ X + c . To this end, ﬁx continuity points x < y < z of F X ( · ) and let ± > 0 be small enough for x + ± < y < z - ± . Then, the monotonicity of P yields that, P ( X n x ) - P ( Y n c + ± ) P ( X n + Y n y + c ) P ( X n z ) + P ( Y n c - ± ) . Since X n D -→ X , taking n → ∞ gives P ( X x ) lim inf n →∞ P ( X n + Y n y + c ) lim sup n →∞ P ( X n + Y n y + c ) P ( X z ) Since continuity points of F X are a dense subset of R , we can let x y and z y , deducing out of the continuity of F X ( · ) at y that P ( X n + Y n y + c ) P ( X y ) when n → ∞ . This applies at every continuity point y of F X ( · ), or equivalently, at every continuity point y + c of F X + c ( · ), as stated. (b). Suppose now that Y n = Z n - X n D -→ 0. Then, by the preceding proof, X n D -→ X implies that Z n = X n + Y n D -→ X . Further, since Y n D -→ 0, also - Y n D -→ 0, hence Z n D -→ X implies that X n = Z n - Y n D -→ X , and we are done. (c). With X n D -→ X if follows from part (b) of the Portmanteau theorem that lim sup n →∞ P ( | X n | ≥ M ) P ( | X | ≥ M ) 0 as M ↑ ∞ . That is, for any δ > 0 there exists M < such that sup n 1 P ( | X n | ≥ M ) δ . Since Y is non- random, clearly Y X n D -→ Y X and view of part (b), it suﬃces to prove that V n X n D -→ 0 for V n = Y n - Y . Our assumption that

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hw7sol - Stat 310A/Math 230A Theory of Probability Homework...

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