Weak law forces a characteristic-function derivative ($S_n/n\to a\text{ in probability}⟹{\varphi }^{\prime }(0)=ia$)

For independent and identically distributed random variables with characteristic function $\varphi$, suppose their sample mean converges in probability to a finite constant $a$. Bounded exponential tests give $e^{-iau}\varphi (u/n{)}^n\to 1$ 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 $n\log \varphi (u/n)-iau\to 0$ uniformly; an integer-valued branch difference is continuous and therefore zero. For arbitrary $h\to 0$, put $n=\lfloor 1/|h|\rfloor$ and $u=nh$, so $1/2\le |u|\le 1$. Dividing the uniform limit by $u$ proves $\log \varphi (h)/h\to ia$, and thus ${\varphi }^{\prime }(0)=ia$. No absolute first moment is assumed.