Let $X_1,\dots,X_n$ be iid random variables with mean $\theta$ and variance 1. Let $F_n$ denote the distribution of $\sqrt{n}(\bar{X}-\theta)$, where $\bar{X}=\frac{1}{n}\sum_{i=1}^nX_i$. Let $\Phi$ denote the standard normal distribution. Are there sufficient conditions on the distribution of the $X_i$s so that
$$\sup_g \left| \frac{\int g(x) dF_n(x)}{\int g(x) d\Phi(x)} - 1 \right| \to 0,$$
where the supremum is over all bounded functions on the interval $[0,1]$ (excluding the case $g=0$ $\Phi$-a.s.). Convergence in total variation seems not strong enough.