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Unformatted text preview: a . Prove that there is a ﬁnite constant c such that lim sup n →∞ X n log n = c a.s., and ﬁnd this constant. 2 3. Let X 1 ,X 2 ,... be iid random variables with P ( X 1 > x ) = 1 x 1 / 2 for x ≥ 1, and let M n = max { X 1 ,X 2 ,...,X n } . Prove that M n /n 2 converges in distribution, and ﬁnd the limiting distribution. 3 4. Let X 1 ,X 2 ,... be iid random variables with mean zero and variance one, and let Y n = 2 n X k =1 X k v u u t n X k =1 X 2 2 k . Prove that Y n converges in distribution, and identify the limiting law. 4...
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 Spring '11
 NA
 Probability

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