Introduction to Probability · Problem 8.1.16

Law of Large Numbers — Grinstead & Snell

Mathematics — Probability · weak-vs-strong-law

Problem

In this exercise, we shall construct an example of a sequence of random variables that satisfies the weak law of large numbers, but not the strong law. The distribution of XiX_i will have to depend on ii, because otherwise both laws would be satisfied. (This problem was communicated to us by David Maslen.) Suppose we have an infinite sequence of mutually independent events A1,A2,…A_1, A_2, \ldots. Let ai=P(Ai)a_i = P(A_i), and let rr be a positive integer. (a) Find an expression of the probability that none of the AiA_i with i>ri > r occur. (b) Use the fact that x−1≤e−xx - 1 \leq e^{-x} to show that P(No Ai with i>r occurs)≤e−∑i=r∞aiP(\text{No } A_i \text{ with } i > r \text{ occurs}) \leq e^{-\sum_{i=r}^{\infty} a_i} (c) (The first Borel-Cantelli lemma) Prove that if ∑i=1∞ai\sum_{i=1}^{\infty} a_i diverges, then P(infinitely many Ai occur)=1.P(\text{infinitely many } A_i \text{ occur}) = 1. Now, let XiX_i be a sequence of mutually independent random variables such that for each positive integer i≥2i \geq 2, P(Xi=i)=12ilog⁡i,P(Xi=−i)=12ilog⁡i,P(Xi=0)=1−1ilog⁡i.P(X_i = i) = \frac{1}{2i\log i}, \quad P(X_i = -i) = \frac{1}{2i\log i}, \quad P(X_i = 0) = 1 - \frac{1}{i\log i}. When i=1i = 1 we let Xi=0X_i = 0 with probability 1. As usual we let Sn=X1+⋯+XnS_n = X_1 + \cdots + X_n. Note that the mean of each XiX_i is 0. (d) Find the variance of SnS_n. (e) Show that the sequence ⟨Xi⟩\langle X_i \rangle satisfies the Weak Law of Large Numbers, i.e. prove that for any ϵ>0\epsilon > 0 P(∣Snn∣≥ϵ)→0,P\left(\left|\frac{S_n}{n}\right| \geq \epsilon\right) \to 0, as nn tends to infinity. We now show that {Xi}\{X_i\} does not satisfy the Strong Law of Large Numbers. Suppose that Sn/n→0S_n/n \to 0. Then because Xnn=Snn−n−1nSn−1n−1,\frac{X_n}{n} = \frac{S_n}{n} - \frac{n-1}{n}\frac{S_{n-1}}{n-1}, we know that Xn/n→0X_n/n \to 0. From the definition of limits, we conclude that the inequality ∣Xi∣≥12i|X_i| \geq \frac{1}{2}i can only be true for finitely many ii. (f) Let AiA_i be the event ∣Xi∣≥12i|X_i| \geq \frac{1}{2}i. Find P(Ai)P(A_i). Show that ∑i=1∞P(Ai)\sum_{i=1}^{\infty} P(A_i) diverges (use the Integral Test). (g) Prove that AiA_i occurs for infinitely many ii. (h) Prove that P(Snn→0)=0,P\left(\frac{S_n}{n} \to 0\right) = 0, and hence that the Strong Law of Large Numbers fails for the sequence {Xi}\{X_i\}.

Related theory

Browse the problem library