Consider a family of distributions $\mathcal{P}=\{P_{\theta}: \theta \in \mathbb{R}^d\}$, with identifiable parameterization. If a sequence of estimators $\hat{\theta}_n \overset{a.s.}{\to} \theta_0$, does this necessarily imply a.s. convergence of the characteristic functions? If not, what would be the sufficient conditions? i.e., let $X_n$ denote a random variable following $P_{\hat{\theta}_n}$, and $X$ following $P_{\theta_0}$, what's the condition for a.s. convergence of $\varphi_{X_n}(t) \to \varphi_X(t)$?
Or more specifically, what I'm trying to get at is, whether $\hat{\theta}_n \overset{a.s.}{\to} \theta_0$ implies a.s. convergence of the cdf.
And what if $\hat{\theta}_n \overset{p}{\to} \theta_0$?
The question I'm trying to answer here is, suppose I have a parametric family in mind for some i.i.d. data $X_1, \dots, X_n \sim P_{\theta_0}$, where $\theta_0$ denotes the true unknown parameter. Say I have a sequence of estimators that's consistent, i.e., $\hat{\theta}_n \overset{p}{\to} \theta_0$, or even stronger, $\hat{\theta}_n \overset{a.s.}{\to} \theta_0$. I want to understand under what conditions I can get uniform convergence of the cdf, i.e., $$\sup_{x} |F_{\theta_0}(x) - F_{\hat{\theta}_n}(x)| \to 0.$$
I know that if the cdf is a continuous function of $\theta$, I'll get $F_{\hat{\theta}_n}(x) \overset{a.s.}{\to} F_{\theta_0}(x)$ for all $x$ by the continuous mapping theorem. And from there, using the same proof of the Glivenko Cantelli theorem, I'll get uniform convergence. But I wonder whether there are milder conditions that could make it work.
