This question is related to a previous question I asked: Using the Uniform Continuity of the Characteristic Function to Show it's Differentiable
Suppose we have a characteristic function $\varphi$ of i.i.d. $X_{i}$. I know that for all $t$, $$ n(\varphi(t/n)-1)\to iat\qquad\qquad\qquad(*) $$ pointwise as $n\to \infty$. To solve my problem, I need to prove that this convergence is uniform in $t$ in some small compact interval around $1$; say $[1-\delta,1+\delta]$. Using the fact that a family of characteristic functions is equicontinuous, and the fact that $S_{n}/n\to a$ weakly, where $S_{n}=X_{1}+\cdots+X_{n}$, I was able to prove that $(\varphi(t/n))^{n} \to e^{iat}$ uniformly in $t$ on compact sets. I also know that characteristic functions are uniformly continuous. But, I am unsure how to put these pieces together to obtain that the convergence $(*)$ is uniform in $t$ in some compact interval centred at $1$.
Any help is greatly appreciated.
I am also open to other ways to solve this problem.
