Characteristic-function derivative under an absolute first moment ($\varphi \prime (0)=i\mathbb{E}X$)

If $\mathbb{E}|X|<\infty$, the characteristic function $\varphi (u)=\mathbb{E}{e}^{iuX}$ satisfies ${\varphi }^{\prime }(0)=i\mathbb{E}X$. The bound $|(e^{iuX}-1)/u|\le |X|$ and the dominated convergence theorem justify differentiation at zero. For mean zero, $\psi (u)=|1-\varphi (u)|/|u|$ extends continuously by $\psi (0)=0$, and $|1-\varphi (u{)}^n|\le n|u|\psi (u)$ by a finite geometric series. This supplies a direct characteristic-function proof of the weak law of large numbers.

// Target: martingale.bigb