Suppose that we are interested in a parameter (or functional) $\theta$ attached to the distribution of observations $\left\{ X_{1},\ldots,X_{n}\right\} $. A popular method for finding an estimator $\hat{\theta}_{n}=\hat{\theta}_{n}(X_{1},\ldots,X_{n})$ is to maximize a criterion function of the type \begin{equation} \theta\mapsto M_{n}(\theta)=\frac{1}{n}\sum_{i=1}^{n}m_{\theta}(X_{i}) \end{equation} Here $m_{\theta}:\mathcal{X}\mapsto\overline{\mathbb{R}}$ are known functions. An estimator maximizing $M_{n}(\theta)$ over $\Theta$ is called an M-estimator
\begin{equation} \hat{\theta}_{n}=\arg\max_{\theta\in\Theta}\frac{1}{n}\sum_{i=1}^{n}m_{\theta}(X_{i}) \end{equation}
There are a lot investigation on the asymptotic behavior of sequences of M-estimators \begin{equation} \sqrt{n}\left(\hat{\theta}_{n}-\theta\right)\to^{D}... \end{equation} where \begin{equation} \theta=\arg\max_{\theta\in\Theta}E_{\theta}m_{\theta}(X). \end{equation}
I want to study the asymptotic distribution of the criterion function, i.e. \begin{equation} \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}m_{\hat{\theta}_{n}}(X_{i})-E_{\theta}m_{\theta}(X)\right)\to^{D}? \end{equation} or \begin{equation} \sqrt{n}\left(M_{n}(\hat{\theta}_{n})-M(\theta)\right)\to^{D}? \end{equation}
I read several books, e.g., Asymptotic Statistics by van der Vaart, but find no comment on this issue.
Note that one can not solve this issue by the Delta theorem, because $M_{n}(\hat{\theta}_{n})$ depends on the observations, thus it is not a deterministic function of $\theta$.