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 nγ ( S nES 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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 Spring '09
 Reed
 Probability, Probability theory, Xn

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