# Convergence example

I'm studying convergence in my probability class and I'm asked to show if there exists any convergence for the following sequence of random variables:

$$\left\{\frac{W_n}{ln(n)}\right\}_{n\geq1} \ s.t. W_n\sim exp(1)$$

I have been able to show that this sequence converges to $$0$$ in probability by Markov inequality, but I'm struggling to prove if there is almost sure convergence to $$0$$ in this case. I know I'm assumed fo use Borel Cantelli lemma, and my specific doubt is if it is right to assume that the right set for which to apply the lemma is:

$$A_n=\left\{\frac{W_n}{ln(n)}=0\right\}$$

If so, I guess $$\sum P(A_n)=0$$ because $$W_n$$ has probability zero of assuming a point as it is a continuous distribution. I thought this is right but I'm only suspicious I did something wrong with this logic because $$\lim \frac{W_n}{ln(n)}$$ could become a different random variable that has probability 1 of assuming the value 0, but I dont know how to show that if it is the case.

If anyone could help me with this, I'd be very grateful.

• If the $X_n$ are independent you want the Borel-Cantelli lemma: en.wikipedia.org/wiki/… Jun 23, 2020 at 3:44
• Okay, I understand why I should use the Borel Cantelli lemma, I'll edit to be more specific Jun 23, 2020 at 3:46
• @ThomasLumley Now I explained my doubts with a little bit more details Jun 23, 2020 at 3:55

Fix $$\epsilon>0$$ and consider the event $$A_n = \{X_n > \epsilon\log n\}$$. If we write $$Z_n$$ for your sequence $$X_n/\log n$$, then $$A_n$$ is the event $$\{Z_n>\epsilon\}$$.
So when $$P(\textrm{A_n infinitely often})=0$$ for all $$\epsilon>0$$ you have $$Z_n\stackrel{a.s.}{\to} 0$$ Conversely, if there is an $$\epsilon>0$$ such that $$P(\textrm{A_n infinitely often})\neq 0$$ then you don't have almost sure convergence.