# Math Help - Almost sure convergence

1. ## Almost sure convergence

Does $(X_{n})_{n_{\geq 1}}$, $X_{n}=\frac{n}{ln(n)}Y_{n}$, where $Y_{n} \sim Exp(n)$ converge a.s. to 0?

2. Originally Posted by yavanna
Does $(X_{n})_{n_{\geq 1}}$, $X_{n}=\frac{n}{ln(n)}Y_{n}$, where $Y_{n} \sim Exp(n)$ converge a.s. to 0?
You should use Borel-Cantelli lemma. Have you tried?

3. $\sum_{n=1}^{\infty}P(nY_n\ln n>\epsilon) =\sum_{n=1}^{\infty}P(Y_n>\epsilon \ln n/n)$

$=\sum_{n=1}^{\infty}\int_{\epsilon \ln n/n}^{\infty}{e^{-x/n}dx\over n}$

$=\epsilon \sum_{n=1}^{\infty}{1\over n^2}<\infty$ for all $\epsilon >0$