0
$\begingroup$

Here's the problem transcribed from the book:

Use the Borel Cantelli lemma to prove that given any sequence of random variables $\{X_n, n \ge 1\}$ whose range is the real line, there exist constants $c_n \to \infty$ such that $$ P[\lim_{n \to \infty} \frac{X_n}{c_n}=0] = 1. $$

Give a careful description of how you choose $c_n$.

Basically I get confused when I think about picking a $c_n$ such that $\sum P(\frac{X_n}{c_n} \ge \epsilon) < \infty$.

$\endgroup$
  • $\begingroup$ There seems to be a typo. Is it X_n/c_n instead of a_n/b_n ? $\endgroup$ – madprob Oct 2 '12 at 1:46
  • $\begingroup$ right, that's true. sorry $\endgroup$ – Taylor Oct 2 '12 at 1:48
4
$\begingroup$

Do you know a set of constants $b_n$ such that $P(|X_n| > b_n) \leq \frac{1}{n^2}$?

Think about the quantiles of $|X_n|$.

Now, can you choose $c_n$ s.t. $P(\frac{|X_n|}{|c_n|} > \epsilon) = P(|X_n| > b_n)$?

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.