91Ó°ÊÓ

Let \(X_{1}, \ldots, X_{n}\) be independent real random variables and let \(S_{k}=\) \(X_{1}+\ldots+X_{k}\) for \(k=1, \ldots, n\). Show that for \(t>0\) Etemadi's inequality holds: $$ \mathbf{P}\left[\max _{k=1, \ldots, n}\left|S_{k}\right| \geq t\right] \leq 3 \max _{k=1, \ldots, n} \mathbf{P}\left[\left|S_{k}\right| \geq t / 3\right]. $$

Show the following improvement of Theorem 5.16: If \(X_{1}, X_{2}, \ldots\) \(\in \mathcal{L}^{2}(\mathbf{P})\) are pairwise independent with bounded variances, then \(\left(X_{n}\right)_{n \in \mathbb{N}}\) fulfils the strong law of large numbers.