(very hard) 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. As a
preliminary, we need to prove a lemma, which is one of the Borel-Cantelli
lemmas.

Suppose we have an infinite sequence of mutually independent events
.
Let
,
and let *r* be a positive integer.

- a.
- Find an expression for the probability that none of the
*A*_{i}with*i*>*r*occur. - b.
- Use the fact that
to show that

- c.
- Prove that if
diverges, then

When

- d.
- Find the variance of
*S*_{n}. - e.
- Show that the sequence
satisfies the weak
law of large numbers. I.e. prove that for any

as*n*tends to infinity.

we know that . From the definition of limits, we conclude that the inequality can only be true for finitely many

- f.
- Let
*A*_{i}be the event . Find . Show that diverges (think integral test). - g.
- Prove that
*A*_{i}occurs for infinitely many*i*. - h.
- Prove that

and hence that the strong law of large numbers fails for the sequence .