Given $X_1,...,X_n$ i.i.d. with unknown variance $\sigma^2$ and
$S^2 = \frac{1}{n-1}\sum_{i =1}^{n}(X_i - \bar X)^2$ the sample variance
we know that $S^2$ converges strongly and therefore in probability to to the unknown variance $\sigma^2$ and that the convergence rate is (at least) $O\left(\frac{1}{n}\right)$ (can be shown with the inequality of Tschebycheff).
Can we say something about the convergence rate of the reciprocal? I.e., what is the convergence rate of $\frac{1}{S^2}$ to $\frac{1}{\sigma^2}$?