Theorem (Weak law of large numbers): Let $\x, \x_1, \x_2, \ldots$ be independently and identically distributed random vectors such that $\Ex\norm{\x} < \infty$. Then $\bar{\x}_n \inP \bmu$, where $\bmu = \Ex(\x)$.
Proof: Let $\varphi$ denote the characteristic function of $\x$. The numbers in parentheses to the side refer to the numbered properties of characteristic functions.
\[\begin{alignat*}{2} \varphi_{\bar{\x}}(\t) &= \varphi_{\Sigma \x_i}(\t/n) &\hspace{8em}& (2) \\ &= \varphi (\t/n)^n && (4); \{\x_i\} \textnormal{ are independent} \\ &= [\varphi(\zero) + \nabla\varphi(\tilde{\t}/n) \tp \t/n ]^n && \href{taylor-expansion.html}{\textnormal{Taylor expansion}}; (5); \Ex\norm{\x} < \infty \\ &= [1 + \nabla\varphi(\tilde{\t}/n) \tp \t/n ]^n && (1) \\ &\to \exp(i\bmu \tp \t) && \href{exponential-limit.html}{\textnormal{Exponential limit}}; (5) \\ &= \varphi_{\bmu} && (7) \\ \bar{\x} &\inD \bmu && \href{characteristic-function.html}{\textnormal{Levy continuity}} \\ \bar{\x} &\inP \bmu && \href{weak-convergence-theorem.html}{\textnormal{Weak convergence}} \end{alignat*}\]