Problem: Let $\xi_1,\xi_2,...$ be a sequence of independent ientically distributed random variables. Show that $E \sup_n |\frac{\xi_n}{n}|<\infty \Leftrightarrow E|\xi_1| \log^+ |\xi_1|<\infty$.
This is a problem from Shiryaev's Probability Theory.
Kolmogorov's Law of Large numbers: If $\xi_1,\xi_2,...$ be independent identically distributed random variables, then $E |\xi_1|<\infty \Leftrightarrow n^{-1} S_n \rightarrow E \xi_1$ and $E |\xi_1|=\infty \Leftrightarrow \lim \sup n^{-1} S_n=+\infty$.
Back to the problem, for $\Rightarrow$, $E |\xi_1| \log^+ |\xi_1|<\infty$, we get $E |\xi_1|<\infty$. From Kolmogorov's law of large number, we get $\frac{\xi_n}{n} \rightarrow E \xi_1$. So $\frac{\xi_n}{n}=\frac{S_n-S_{n-1}}{n}=E\xi_1-E\xi_1=0$. So $P(|\xi_n|>n i.o)=0$ and $\sup_n |\frac{\xi_n}{n}| \leq 1$, and $E \sup_n |\frac{\xi_n}{n}|<\infty$.
For $\Leftarrow$,I am stuck here. I think the approach is based on Kolmogorov's Law of Large number again, but I am not sure how to apply $E|\xi_1|<\infty$ to get $E|\xi_1| \log^+ |\xi_1|<\infty$, and how to apply $E \sup_n |\frac{\xi_n}{n}|<\infty$ to get $\frac{S_n}{n}$ to converge. Thanks in advance!