Probability2010jan

# Probability2010jan - a Prove that there is a ﬁnite...

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PhD Qualifying Exam, January 2010 Probability You may use the following facts. (i) Y is Poisson with parameter λ > 0 if P ( Y = k ) = e - λ λ k /k ! for k = 0 , 1 ,... , in which case E ( Y ) = λ and V ar ( Y ) = λ . (ii) Y is exponential with parameter λ > 0 if Y has density function f ( t ) = λe - λt for t > 0 and f ( t ) = 0 for t 0, in which case E ( Y ) = 1 and V ar ( Y ) = 1 2 . (iii) If U n converges to a constant c a.s. as n → ∞ , and V n converges weakly to V , then U n V n converges weakly to cV . 1. Let a 1 ,a 2 ,... be a sequence of positive numbers such that a n = . Let X 1 ,X 2 ,... be independent random variables such that for each n , X n is Poisson with parameter a n . Prove that for S n = X 1 + ··· + X n , S n /E ( S n ) 1 in probability as n → ∞ . 1

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2. Fix a > 0, and let X 1 ,X 2 ,... be iid random variables which are exponential with parameter

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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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Probability2010jan - a Prove that there is a ﬁnite...

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