Grinstead-Snell-Probability · Problem 16

Law of Large Numbers

Mathematics · weak-vs-strong-law-large-numbers

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

X_i

will have to depend on

i

, because otherwise both laws would be satisfied. Suppose we have an infinite sequence of mutually independent events

A_1, A_2, \ldots

. Let

a_i = P(A_i)

, and let

r

be a positive integer. (a) Find an expression of the probability that none of the

A_i

with

i > r

occur. (b) Use the fact that

x - 1 \leq e^{-x}

to show that

P(\text{No } A_i \text{ with } i > r \text{ occurs}) \leq e^{-\sum_{i=r}^{\infty} a_i}

\sum_{i=1}^{\infty} a_i

diverges, then

P(\text{infinitely many } A_i \text{ occur}) = 1.

Now, let

X_i

be a sequence of mutually independent random variables such that for each positive integer

i \geq 2

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 = 1

we let

X_i = 0

with probability 1. As usual we let

S_n = X_1 + \cdots + X_n

. Note that the mean of each

X_i

is 0. (d) Find the variance of

S_n

. (e) Show that the sequence

\langle X_i \rangle

satisfies the Weak Law of Large Numbers, i.e. prove that for any

\epsilon > 0

P\left(\left|\frac{S_n}{n}\right| \geq \epsilon\right) \to 0,

n

tends to infinity. (f) Let

A_i

be the event

|X_i| \geq \frac{1}{2}i

. Find

P(A_i)

. Show that

\sum_{i=1}^{\infty} P(A_i)

diverges (use the Integral Test). (g) Prove that

A_i

occurs for infinitely many

i

. (h) Prove that

P\left(\frac{S_n}{n} \to 0\right) = 0,

and hence that the Strong Law of Large Numbers fails for the sequence

\{X_i\}

Related theory

The Strong Law Of Large Numbers

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