I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim):
If $X_n$ is ANY sequence of random variables, there are constants $c_n \to \infty$ so that $$\frac{X_n}{c_n} \to 0 \quad \mbox{a.s}$$
You will find it in Chapter 2: Laws of Large Numbers, under the section on Borel Cantelli Lemma. What makes this question unusual is that there are NO assumptions made on the random variables.
My attempt: I rephrased the question equivalently as:
If $X_n$ is ANY sequence of random variables, there are constants $a_n \to 0$ so that we need to show $$a_nX_n \to 0 \quad \mbox{a.s}$$
Then I showed that it was sufficient to assume $a_n > 0$ and $X_n \geq 0$. To see this, since the limit is going to , the sign of the constants do not matter, hence positivity of $a_n$ can be assumed w.l.o.g. As for an arbitrary r.v, any r.v can be written as:
$$X_n = X_n^+ - X_n^-$$
where $X_n^+ = \max(X_n,0)$ and $X_n^- = \max(-X_n,0)$
Suppose we prove the result for non-negative random variables, then say we have $$b_n X_n^+ \to 0 \quad c_n X_n^- \to 0 \quad \mbox{a.s}$$
Then pick $a_n = \min(b_n,c_n)$. Then this will ensure $$a_nX_n \to 0 \quad \mbox{a.s}$$
For the non-negative case, I was able to prove the result for simple functions: If $X_n$ is simple, and $X_n = \sum_{k=1}^n s_k 1_{A_k}$, take $a_n = \frac{1}{2^{n} \sum_{k=1}^n s_k}$ would work, but only if all functions were simple.
Now the tough part. Handling the non -ve measurable function case. I had difficulty here.
Hence my question is:
I'd like a hint/answer (preferably a hint) on how to solve this particular case. Right now I am looking at manipulating the lemma that every non-negative measurable function can be approximated by a monotone sequence of simple functions.
Thank you.
Note: I use measurable functions and random variables interchangeably. But note that the space is a probability space. Additionally, I didn't find any similar question (I typed convergence random variables). If it has been answered, kindly provide the link.