I am looking for assymptotically tight bounds on \begin{align} P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]=P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} \right| \le \frac{1}{n} \right] \end{align} where $Z_i$ are i.i.d. standard normal.
I am sure these are available in the literature. I will Be grateful for a reference.
I am ultimately interested in
\begin{align} \lim_{n \to \infty} P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} \right| \le \frac{1}{n} \right]=??? \end{align}
Obviously, if we replace $\frac{1}{n}$ by some $\epsilon$ by a strong law of large number we would have that the limit is zero. However, here things depended on the $n$.
By central limit theorem $\frac{\sum_{i=1}^n Z_i^2 }{n}$ is distributed as $N(1, \frac{2}n)$ for large $n$. Then your probability is:
$$P \approx \int_{1-\frac{1}n}^{1+\frac{1}n}\frac{\sqrt{n}}{\sqrt{4\pi}}e^{-\frac{n(x-1)^2}{4}}dx \approx \frac{1}{\sqrt{\pi n}}$$