= Weak law forces a characteristic-function derivative
{title2=$S_n/n\to a\text{ in probability}\;\Longrightarrow\;\phi'(0)=ia$}
For <independent and identically distributed random variables> with <characteristic function> $\phi$, suppose their <sample mean> converges in probability to a finite constant $a$. Bounded exponential tests give $e^{-iau}\phi(u/n)^n\to1$ uniformly on bounded $u$ intervals: the error is at most $T\delta+2\mathbb P(|S_n/n-a|>\delta)$ on $|u|\le T$. Taking the continuous <complex logarithm> vanishing at zero shows $n\log\phi(u/n)-iau\to0$ uniformly; an integer-valued branch difference is continuous and therefore zero. For arbitrary $h\to0$, put $n=\lfloor1/|h|\rfloor$ and $u=nh$, so $1/2\le|u|\le1$. Dividing the uniform limit by $u$ proves $\log\phi(h)/h\to ia$, and thus $\phi'(0)=ia$. No absolute first moment is assumed.
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