831_09hw3

831_09hw3 - 831 Theory of Probability Fall 2009 Homework 3...

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Unformatted text preview: 831 Theory of Probability Fall 2009 Homework 3 Due Thursday, October 1 1. Let θ > 0. Show that, for a > 0, lim e−nθ (nθ)k k! k : 0≤k≤na = 0, if a < θ 1, if a > θ. n→∞ Hint: apply the WLLN to Poisson random variables. 2. For each n ∈ N let {Xn,k : 1 ≤ k ≤ n} be IID random variables such that 0 ≤ Xn,k ≤ C (same constant C for all n and k ). Let Sn = Xn,1 + Xn,2 + · · · + Xn,n . Assume that ESn → ∞ and Var Sn → ∞. Show that Sn → ∞ in probability. The natural way to interpret this statement is that for any k < ∞, P {Sn ≤ k } → 0 as n → ∞. Hint: since ESn → ∞, try to control the distance of Sn to its mean with Chebychev’s inequality. 3. Let {pn } be given numbers such that 0 ≤ pn ≤ 1. Let {Xn } be independent random variables with distributions P (Xn = 1) = 1 − P (Xn = 0) = pn . (a) Find the condition on {pn } that is equivalent to Xn → 0 almost surely. Find the condition on {pn } that is equivalent to Xn → 0 in probability. Give an example where Xn → 0 in probability but not almost surely. (b) On the sample space Ω = [0, 1] with P = Lebesgue measure, let Yn (ω ) = 1[0,pn ] (ω ). Show that, even if {pn } satisfies P (Yn = 1) = ∞, the conclusion of the second Borel-Cantelli Lemma can fail. Why is that? 4. Exercise 6.17 from p. 54 of Durrett. That is, show that in the St. Petersburg game, lim Sn /(n log2 n) = ∞ a.s., in contrast with the weak law n→∞ we discovered. ...
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