This post is a follow-up to my post on 12/29/2023, where I posed the following question:
Let
be an i.i.d. sequence of
-valued random variables. Let
be a positive sequence such that
Under what condition of
can one conclude that
almost surely?
I have found a full characterization of the sequence After all, it is an interesting exercise.
Proposition: Let
be an i.i.d. sequence of
-valued random variables Let
be a positive sequence such that
If
for any constant
then
almost surely. Otherwise, almost surely
does not converge.
As an application, if is an i.i.d. sequence of mean-one exponentially distributed random variables, then almost surely the sequence of
does not converge.
Proof: First, suppose that there exists such that
Then the independent events
satisfies
By Borel-Cantelli’s Lemma, almost surely the events occurs for infinitely many
Therefore, with probability 1, the sequence
does not converge to 0. Now fix
We show that almost surely
. Consider the independent events
One has
By Borel-Cantelli’s Lemma, almost surely the events occurs for infinitely many
Therefore, with probability 1, the sequence
does not converge to
Next, suppose that for every , one has
For a fixed constant
, the event
is a tail event. Thus,
or
according to Kolmogorov’s 0-1 Law. Note that
and
which is positive because Thus,
and therefore,
Now note that
This completes the proof.