Let $\{\mu_n:n\in\mathbb N\}$ and $\mu$ be distributions on $\mathbb R$, and let $\{\phi_n:n\in\mathbb N\}$ and $\phi$ be their respective characteristic functions.
We can easily show using a direct application of the so called continuous mapping theorem that if $\mu_n\to\mu$ weakly, then $\phi_n\to\phi$ pointwise. I also know that if $\mu_n\to\mu$ weakly,
then for any bounded set $B\subset\mathbb R$,
we have that $\phi_n\to\phi$ uniformly. However, my attempts to understand why this is true have so far been in vain.
I've tried to find upper bounds for
\begin{align*}
\sup_{t\in[a,b]}\left|\int_{-\infty}^{\infty}e^{itx}~\mu_n(dx)-\int_{-\infty}^{\infty}e^{itx}~\mu(dx)\right|,
\end{align*}
but it always ends up being too small or too big. Any hint would be greatly appreciated.
Best Answer
Hints