Without using the Central Limit Theorem or the Law of the Iterated Logarithm, prove the Weak Law of Large Numbers when \(X_{i} \in L^{1}\), i.e. show that when \(X_{1}, X_{2}, \ldots \) is a sequence of i.i.d. variables with \(\mathbb {E}[X_{1}] = \mu \) and \(X_{1} \in L^{1}\), then
\[ \bar {X} = \frac {X_{1} + \cdots + X_{n}}{n} \to \mu \quad \text {in } L^{1}. \]
Hint: first, prove the claim when \(X_{1} \in L^{2}\). Then for \(a > 0\), consider
\[ X_{i}^{a} = X_{i} \cdot \mathbf {1}\{|X_{i}| \le a\}. \]