Let $X_1,...,X_n$ be iid observations from $N(0,1)$. Let $\overline{X}=\dfrac{1}{n}\sum_{i=1}^n X_i$ and $S^2=\dfrac{1}{n}\sum_{i=1}^n (X_i-\overline{X})^2$. Then is it true that $\sqrt{n}\sup_x |\Phi(\dfrac{x-\overline{X}}{S})-\Phi(x)|\stackrel{p}{\to} 0$? Here $\Phi(.)$ is standard normal cdf.
Seems something like Berry Esseen bound but I think I am getting it is stochastically bounded and not really going to 0.
No, it is not true. First fix any $x$ and consider $\overline{X^2}=\frac1n\sum_{i=1}^nX_i^2$, $S^2=\overline{X^2}-\bigl(\overline X\bigr)^2$ and $h(s,t)=\Phi\left(\dfrac{x-s}{\sqrt{t-s^2}}\right)$. Then use multivariate Delta method to prove that $$ \sqrt{n} \left(\Phi\biggl(\dfrac{x-\overline{X}}{S}\biggr)-\Phi(x)\right)=\sqrt{n}\biggl(h\biggl(\overline X, \overline{X^2}\biggr)-\underbrace{h(\mathbb EX_1,\mathbb EX_1^2)}_{h(0,1)}\biggr)\xrightarrow{\mathcal D}\mathcal N(0,\sigma_x^2), $$ where $\sigma^2_x = \nabla h(0,1)^T\cdot \Sigma \cdot \nabla h(0,1)$ and $\Sigma$ is a covariance matrix of $(X_1,X_1^2)$.
Here $\text{Var}(X_1)=1$, $\text{Var}(X_1^2)=2$, $\text{Cov}(X_1, X_1^2)=0$, so $$ \Sigma=\begin{pmatrix}1 & 0 \cr 0 & 2 \end{pmatrix} $$ $$ \frac{\partial h}{\partial s}\bigg|_{s=0, t=1} = -\varphi(x) = -\frac{1}{\sqrt{2\pi}}e^{-x^2/2} $$ and $$ \frac{\partial h}{\partial t}\bigg|_{s=0, t=1} = -\frac12\varphi(x) = -\frac{1}{2\sqrt{2\pi}}e^{-x^2/2} $$ Please check the derivatives, although their values do not matter much. And $$ \sigma^2_x = \varphi^2(x)\cdot 1+\left(\frac{\varphi(x)}{2}\right)^2\cdot 2 = \frac32\varphi^2(x). $$ Say, for $x=0$, $$ \sqrt{n} \left(\Phi\biggl(\dfrac{x-\overline{X}}{S}\biggr)-\Phi(x)\right) \xrightarrow{\mathcal D} \mathcal N\biggl(0,\frac{3}{4\pi}\biggr) $$ For any $\varepsilon>0$ $$ \liminf_{n\to\infty}\mathbb P\left(\sqrt{n}\sup_x \biggl|\Phi\biggl(\dfrac{x-\overline{X}}{S}\biggr)-\Phi(x)\biggr|>\varepsilon\right) \geq \liminf_{n\to\infty}\mathbb P\left(\sqrt{n} \biggl|\Phi\biggl(\dfrac{0-\overline{X}}{S}\biggr)-\Phi(0)\biggr|>\varepsilon\right) $$ $$ =\mathbb P(|Y|>\varepsilon)=2\Phi\left(-\frac{2\sqrt{\pi}\varepsilon}{\sqrt3}\right)>0 $$ where $Y\sim N\bigl(0,\frac{3}{4\pi}\bigr)$.
