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 $\sum P(\frac{X_n}{c_n} \ge \epsilon) < \infty$.

  • 0
    right, that's true. sorry2012-10-02

1 Answers 1

4

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. $P(\frac{|X_n|}{|c_n|} > \epsilon) = P(|X_n| > b_n)$?