Let $X_1, X_2, ...$ denotes a sequence of i.i.d. random variables such that $X_1$ ~ $exp(1)$ and c>0. What is $P( X_n/\log(n) > c$ for infinitely many $n$'s) ?
Can I simply say that $P(X_n > c \cdot \log(n)) = 1/(n^c)$, where $c > 0$; and with the Borel-Cantelli Lemma, the probability is $1$ if $c \le 1$ and $0$ otherwise?