I have been stuck on the following problem:
Let $X_i$ iid non-negative random variables (not only necessarily integer-valued) and define $ A:=\limsup \frac{X_i}{i}$. Prove that $\mathbb{E}X_i<\infty\Rightarrow A=0$ and $\mathbb{E}X_i=\infty\Rightarrow A=\infty$. The hint says that it is an immediate consequence of Borel-Cantelli, and I tried to apply it together with Markov inequality, but that leads nowhere.
Any help/reference would be greatly appreciated.
Suppose $E[X_i]<\infty$, then for all $c>0$, $$ E[X/c] = \int_0^\infty P(X/c>t)\,dt\ge \sum_{k\ge 1}P(X/c>k)= \sum_{k\ge 1}P(X/k>c) $$ Thus, since $\sum P(X/k>c)$ is finite, by first Borel-Cantelli, $P(X/k>c \,\,\,i.o.)=0.$ This holds for all $c>0$, proving $A=0$ a.s. Argument is similar for $E[X_i]=\infty$, but there you use an upper bound on the above integral.