Does uniform convergence of characteristic functions hold on the whole real line?

Question

According to this question uniform convergence of characteristic functions put forward by Shine, we get following result.

Suppose that a sequence of probability measures $\mu_n$ on $(\mathbb{R},\mathfrak{B}_\mathbb{R})$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Then $\phi_n$ converges uniformly to $\phi$ on any compact subset $K\subseteq\mathbb{R}$.

It seems that the uniform convergence of $\phi_n$ to $\phi$ doesn't hold on the whole real line $\mathbb{R}$. Could we find some simple counterexamples? Equivalently, could we find a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $X_n\xrightarrow{D}X_\infty$ but $\{\phi_{X_n}(t)\}$ doesn't uniformly converge to $\phi_{X_\infty}(t)$ on $\mathbb{R}$?

Answer 1

I think $X_n \equiv \frac{1}{n}$ works. They converge to $X_{\infty} \equiv 0$. But $\phi_{X_n}(t) = e^{\frac{it}{n}}$, and $\phi_{X_{\infty}} \equiv 1$. No matter how large $n$ is, we can pick $t$ (e.g. $t = n$) so that $\phi_{X_n}(t)$ is bounded away from $1$.