(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 Xi 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
Suppose we have an infinite sequence of mutually independent events
and let r be a positive integer.
Now, let Xi be a sequence of mutually independent random variables such
that for each positive integer ,
- Find an expression for the probability that none of the
Ai with i > r occur.
- Use the fact that
to show that
- Prove that if
When i=1 we let X1 = 0 with probability 1. As usual we let
Note that the mean of each Xi is 0.
We now show that
does not satisfy the strong law of large
numbers. Suppose that
- Find the variance of Sn.
- 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 i.
- Let Ai be the event
- Prove that Ai occurs for infinitely many i.
- Prove that
and hence that the strong law of large numbers fails for the sequence