I am going over the Borel-Cantelli Problems in Durrett, but I am really stuck on Exercise 2.3.15..
Suppose $Y_{1},Y_{2},\ldots$ are i.i.d.. I need to prove:
1) $\frac{\max _{1\leq m\leq n}Y_{m}}{n}\to 0$ in probability if and only if $nP(Y_{i}>n)\to 0$
2) $\frac{\max _{1\leq m\leq n}Y_{m}}{n}\to 0$ almost surely if and only if $E\left(\max\{0,Y_{i}\}\right)<\infty$.
There are two easier parts to the problem that I was able to solve. I proved that $$ \frac{Y_{n}}{n}\to 0\text{ in probability }\iff P(|Y_{i}|<\infty)=1 $$ and $$ \frac{Y_{n}}{n}\to 0\text{ almost surely }\iff E|Y_{i}|<\infty. $$ However, I am unsure if these facts can be used to solve the above two problems. I'd really appreciate some help with these problems.
Thanks!
The statements you are trying to prove involve the maximum. For the first part, define $P_n:=\mathbb P\left(\max_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon\right)$ for an arbitrary but fixed positive $\varepsilon$. Note that $$P_n=\mathbb P\left(\bigcup_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon\right)=1-\mathbb P\left(\bigcap_{1\leqslant i\leqslant n}Y_i\leqslant n\varepsilon\right)$$ and since the events $\left(Y_i\leqslant n\varepsilon\right)_{1\leqslant i\leqslant n}$ are independent and have the same probability, we derive that $$P_n=1-\left(\mathbb P\left(Y_1\leqslant n\varepsilon\right)\right)^n.$$ The equality $\mathbb P\left(Y_1\leqslant n\varepsilon\right) =\left(1-P_n\right)^{1/n} $ entails $$n\mathbb P\left(Y_1\gt n\varepsilon\right)= n\left(1-\left(1-P_n\right)^{1/n}\right).$$ Now do an asymptotic expansion of $n(1-(1-t)^{1/n})$ to get that $P_n\to 0$ if and only if $n\mathbb P\left(Y_1\gt n\varepsilon\right)\to 0$.
For the second part, one direction is done by the result you mentioned. For the converse, apply the Borel-Cantelli lemma to the sequence of events $(A_n)$ defined by $A_n=\left\{\max_{1\leqslant i\leqslant n}Y_i\gt n\varepsilon \right\}$ (the probability is bounded by $n\mathbb P\left(Y_1\gt n\varepsilon\right))$.