# Berry-Esseen Type For the Unknown Variance

Suppose we have and i.i.d. sample $$X_1,\ldots, X_n$$ with mean $$\mu$$ and finite variance.

Let $$\bar{X}$$ denote the sample mean and $$s_n^2$$ denote that sample variance.

Using classical CLT together with Slutsky's theorem it follows that \begin{align} \frac{\bar{X} -\mu}{ \sqrt{ s_n^2/n}} \to Z \end{align} where $$Z$$ is standard normal and the convergence is in distribution.

Question: Is there a Berry-Esseen type result? That is let $$F_n$$ denote the cdf of $$\frac{\bar{X} -\mu}{ \sqrt{s_n^2/n}}$$, do we have something of the form: \begin{align} \sup_{t \in \mathbb{R}} | F_n(t) -\Phi(t)| \le \frac{B}{\sqrt{n}} \end{align}

I would guess that people have looked into this already, so a reference would be appreciated. Unfortunately, I couldn't find anything myself.

• There have been some papers in recent times relating to such Berry-Esseen-like inequalities in this situation. I don't have any references handy but I might be able to find one. Commented Sep 10, 2022 at 1:07
• @Glen_b Thanks. I would really appreciate it if you do.
– Boby
Commented Sep 10, 2022 at 1:29
• Memory has kicked in. I'm pretty sure at least one of the ones I saw was on arXiv. Looking there now. No luck so far. Commented Sep 10, 2022 at 1:32
• None of these were the papers I saw before ... which were much more recent, but it's a start: projecteuclid.org/journals/annals-of-probability/volume-24/… and link.springer.com/chapter/10.1007/BFb0074828 Commented Sep 10, 2022 at 1:45
• Here's one of the ones I saw before: arxiv.org/abs/1101.3286v2 Commented Sep 10, 2022 at 1:54