I'm looking for some bound on what feels like should be a concentration inequality, e.g. $F(n,\epsilon)$ where
$$\Pr\left(\frac{1}{n}\sum_{i=1}^n \left(X_i-\mathbb E[X]\right)^2 > \epsilon\right) \leq F(n,\epsilon)$$
Here, $X_i$ are i.i.d., and you can assum you know $|X_i| < X_\max$ always.
It's very similar to Bernstein's or Efron Stein, but it isn't. I feel like it should be super simple, but I can't seem to find it. Any pointers would be much appreciated! Thanks!
Edit: It occurs to me that what I'm kind of looking for is some concentration on the variance of $X_i$. Therefore if I could impose this (say, $X_\max^2$). But, what is the concentration of the empirical variance to the true variance?
Edit: Note that I am NOT looking for
$$\Pr\left(\left(\frac{1}{n}\sum_{i=1}^n X_i-\mathbb E[X]\right)^2 > \epsilon\right) \leq F(n,\epsilon)$$
which you can do using a number of different concentration inequalities.
Edit: Ok I think my sleep deprived brain finally figured it out. The term
$$ \frac{1}{n}\sum_{i=1}^n \left(X_i-\mathbb E[X]\right)^2 $$
is exactly the sample variance. Therefore there's no reason to believe it will diminish with large $n$. Therefore there's no real point in bounding it this way.
Nevermind, go ahead and close if you wish :)