There are well-known results about the scaling and centering laws of a sequence of -valued random variables. That is the problem of finding the right coefficients and such that the sequence converges in law. See, for example, Section 14.8 of [1]. Normally, one tries to avoid the degenerate type, which is when the limit distribution is a Dirac mass at 0. I recently ran into a problem where the degenerate type is desirable. More specifically:

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?

This problem seems to be quite classic. If has a bounded range, i.e. for some numbers and , then no additional condition of is needed. How about the case has an unbounded range? This is essentially Problem 51 on page 263 of [1]. The textbook does not ask for a full characterization of the sequence . I find the problem quite interesting.

The first observation is that the event is a tail event. By Kolmogorov’s 0-1 Law, either or . We just need to decide between the two. One may be tempted to think that is essentially the same as , so it should almost surely converge 0 as . This argument is not correct because only has the same distribution as If the range of values of is unbounded, then so is the range of values of . Let us consider two following cases, one of which gives and the other gives Let us denote the cumulative distribution function of by

**Case 1:** there exists a positive sequence such that

and

Let and Then and

which has a finite positive value because It is easy to see that and thus

**Case 2:** there exists a positive sequence such that

and

Let and The events are independent and

By Borel-Cantelli’s Lemma, . It is easy to see that Thus,

**Final thoughts:** Cases 1 and 2 do not cover all possibilities. One can see this through the simple case (the case when the random variables are i.i.d. exponentials of mean one) and It would be interesting to see a full characterization of the sequence for which one has I think this should be known somewhere in the literature.

Update 01/01/2024: I have resolved the problem.

## References:

- “
*A modern approach to Probability Theory*” (1997) by Friestedt and Gray.