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.