$X_1,\ldots,X_n$ are i.i.d standard normal random variables.

$a_1,\ldots, a_n$ are constants with $a_i \in [\kappa_1, \kappa_2]$ for all $i$ and $\kappa_1>0$.

$\hat c_n$ is given as the solution to the equation:

$$\sum_{i=1}^n \frac{c -X_i^2}{(c+a_i)^2}=0. $$

Can we prove that $$ \operatorname{Variance} (\hat c_n)=O(n^{-1}) \text{ as } n \to \infty.$$

Note that the above can be proved easily when all $a_i$' s are equal. What will happen in the general case?

notconvex $\endgroup$ – fedja Jun 4 '17 at 1:41