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831_09hw5

# 831_09hw5 - 1 2,n and τ n k = inf m | Y 1,Y 2,Y m}| = k is...

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831 Theory of Probability Fall 2009 Homework 5 Due Tuesday, October 27 1. Let { X n } , X be real random variables. (a) Suppose X n X in probability. Show that then also X n d X . (b) Suppose X n converges in distribution to a constant c . Show that then X n c also in probability. 2. Let f n ( x ) = 1 - cos(2 πnx ) for n N and x [0 , 1]. Verify that f n is the density of a probability measure μ n on [0 , 1]. Is there a weak limit μ n d μ ? Either show convergence and identify μ or prove that the sequence does not converge. (Sketch f n for large n if you are baffled.) 3. Let S n = X 1 + · · · + X n be a sum of i.i.d. mean zero random variables with a finite variance. Show that lim n →∞ S n / n = a.s. (Hint: CLT and Kolmogorov’s 0-1 law.) 4. Recall the coupon collector’s problem: for a fixed n , Y 1 , Y 2 , Y 3 , . . . are i.i.d. uniform on
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Unformatted text preview: { 1 , 2 ,...,n } , and τ n k = inf { m : |{ Y 1 ,Y 2 ,...,Y m }| = k } is the time when we ﬁrst see k diﬀerent coupons. Fix 0 < θ < 1, and let S n = τ n [ nθ ] , the time by which we have seen fraction θ of the coupons. First ﬁnd the asymptotics of the mean and variance of S n : that is, ﬁnd constants c and σ 2 and exponents α and β such that E ( S n ) ∼ n α c and Var( S n ) ∼ n β σ 2 as n → ∞ . Then prove a central limit theorem: ﬁnd exponent γ such that n-γ ( S n-ES n ) converges weakly to a nondegenerate normal distribution. (It is essential here that θ < 1. a n ∼ b n means a n /b n → 1.)...
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