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 \(\bar{\x}_n \inP \bm\), where \(\bm = \Ex(\x)\).
Proof: Let \(\varphi\) denote the characteristic function of \(\x\). The numbers in parentheses to the side refer to the numbered properties of characteristic functions.
\[\begin{alignat*}{2} \varphi_{\bar{\x}}(\t) &= \varphi_{\Sigma \x_i}(\t/n) &\hspace{8em}& (2) \\ &= \varphi (\t/n)^n && (4); \{\x_i\} \textnormal{ are independent} \\ &= [\varphi(\zero) + \nabla\varphi(\t^*/n) \Tr \t/n ]^n && \href{taylor-series.html}{\textnormal{Taylor series}}; (5); \Ex\norm{\x} < \infty \\ &= [1 + \nabla\varphi(\t^*/n) \Tr \t/n ]^n && (1) \\ &\to \exp(i\bm \Tr \t) && \href{exponential-limit.html}{\textnormal{Exponential limit}}; (5) \\ &= \varphi_{\bm} && (7) \\ \bar{\x} &\inD \bm && \href{continuity-theorem.html}{\textnormal{Continuity theorem}} \\ \bar{\x} &\inP \bm && \href{weak-convergence-theorem.html}{\textnormal{Weak convergence theorem}} \end{alignat*}\]