Degenerate scaling of random variables

There are well-known results about the scaling and centering laws of a sequence of $\mathbb{R}$ -valued random variables. That is the problem of finding the right coefficients $a_n$ and $b_n$ such that the sequence $\frac{X_n-b_n}{a_n}$ 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 $(X_n)$ be an i.i.d. sequence of $\mathbb{R}$ -valued random variables. Let $(a_n)$ be a positive sequence such that $\lim a_n=\infty$ . Under what condition of $(a_n)$ can one conclude that $\lim\frac{X_n}{a_n}=0$ almost surely?

This problem seems to be quite classic. If $X_1$ has a bounded range, i.e. $\mathbb{P}(X_1\in[a,b])=1$ for some numbers $a$ and $b$ , then no additional condition of $(a_n)$ is needed. How about the case $X_1$ 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 $a_n$ . I find the problem quite interesting.

The first observation is that the event $A=\left[\lim\frac{X_n}{a_n}=0\right]$ is a tail event. By Kolmogorov’s 0-1 Law, either $\mathbb{P}(A)=0$ or $\mathbb{P}(A)=1$ . We just need to decide between the two. One may be tempted to think that $\frac{X_n}{a_n}$ is essentially the same as $\frac{X_1}{a_n}$ , so it should almost surely converge 0 as $n\to\infty$ . This argument is not correct because $\frac{X_n}{a_n}$ only has the same distribution as $\frac{X_1}{a_n}.$ If the range of values of $X_1$ is unbounded, then so is the range of values of $\frac{X_1}{a_n}$ . Let us consider two following cases, one of which gives $\mathbb{P}(A)=0$ and the other gives $\mathbb{P}(A)=1.$ Let us denote the cumulative distribution function of $|X_1|$ by $F(x).$

Case 1: there exists a positive sequence $(c_n)$ such that

$\displaystyle \lim\frac{c_n}{a_n}=0$ and $\displaystyle\sum(1-F(c_n))<\infty.$

Let $E_n=[|X_n|\le c_n]$ and $E=\cap_{n=1}^\infty E_n.$ Then $\mathbb{P}(E_n)=F(c_n)$ and

$\displaystyle \mathbb{P}(E)=\prod_{n=1}^\infty F(c_n)=\prod_{n=1}^\infty (1-(1-F(c_n))$

which has a finite positive value because $\sum(1-F(c_n))<\infty.$ It is easy to see that $E\subset A$ and thus $\mathbb{P}(A)=1.$

Case 2: there exists a positive sequence $(c_n)$ such that

$\displaystyle \lim\frac{c_n}{a_n}>0$ and $\displaystyle\sum(1-F(c_n))=\infty.$

Let $E_n=[|X_n|> c_n]$ and $E=\limsup E_n.$ The events $E_n$ are independent and

$\displaystyle \sum \mathbb{P}(E_n)=\sum(1-F(c_n))=\infty.$

By Borel-Cantelli’s Lemma, $\mathbb{P}(E)=1$ . It is easy to see that $E\subset A^c.$ Thus, $\mathbb{P}(A)=1-\mathbb{P}(A^c)=0.$

Final thoughts: Cases 1 and 2 do not cover all possibilities. One can see this through the simple case $F(x)=1-e^{-x}$ (the case when the random variables $X_n$ are i.i.d. exponentials of mean one) and $a_n=2\ln n.$ It would be interesting to see a full characterization of the sequence $(a_n)$ for which one has $\mathbb{P}(A)=1.$ I think this should be known somewhere in the literature.