# Application of Borel-Cantelli theorem to geometric distribution

I have been working on the following question for a while, with not much progress. That would be great if could offer me some help.

Let $(X_n)$ be a sequence of Independence random variables with geometric distribution, $P(X_n=k)=pq^{k-1}$. Let $\beta = (\log{\frac{1}{q}})^{-1}$. Show that with probability 1, $X_n > \beta\log{n}$ for infinitely many n. Show also that if $\alpha > \beta$ then, with probability 1, $X_n > \alpha\log{n}$ for only finitely many n.

For the first part I probably want to define random events $E_n = {X_n > \beta\log{n}}$ and show that $\sum_{i=1}^{\infty}P(E_n)=\infty$. For the second part I am assuming I want to use the other part of Borel-Cantelli theorem, i.e. show that $\sum_{i=1}^{\infty} P(X_n>\alpha\log{n}) <\infty$. I know that $P(X_n>n) = q^n$ but I am not sure how to deal with the logarithms.

Any hints welcome!

Thanks, Szymon

• For every nonnegative real number $x$, $q^{x}\leqslant P(X_n>x)<q^{x-1}$. Can you conclude from this?
– Did
Dec 16, 2016 at 13:31