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 BorelCantelli 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. ...
View
Full Document
 Spring '09
 Reed
 Probability, Probability theory, yn, Sn, Xn, Poisson random variables, iid random variables

Click to edit the document details