Setup
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This question is a followup on [this question][1]. I'm interested in the asymptotic distribution of certain quadratic forms.

So, let $Z$ be a $p$-dimensional random vector with (unknown) distribution $P$. Consider a mapping $g:\mathbb R^q \times \mathbb R^p \rightarrow \mathbb R^k$. Let $\theta^*\in \mathbb R^p$ satisfy $\mathbb E_P[g(Z;\theta^*)] = 0_k$, and let $\theta_n \in \mathbb R^p$ be a $\sqrt{n}$-consistent estimator of $\theta^*$.

Question 1
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Under very general regularity conditions on $g$, what can be said about the asymptotic distribution of the (random) scalar quantity
$$\alpha_n := n\mathbb E_{\hat{P}_n}[g(Z;\theta_n)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta_n)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta_n)]?
$$
Here, $\mathbb E_{\hat{P}_n}[g(Z;\theta)] = (1/n)\sum_{i=1}^ng(z_i;\theta) \in \mathbb R^k$ is the empirical mean of the random vector $g(Z;\theta)$ from an i.i.d sample $z_1,\ldots,z_n\sim P$, and $\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)] \in \mathbb R^{k \times k}$ is the empirical covariance matrix.

Question 2: specialization to MLE
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Same question with $g(Z;\theta) := -\partial_\theta \log p_\theta(Z)$, the score function for a parametric family of distributions (densities thereof) $\{p_\theta\}$, assumed to contain the true distribution of $Z$, namely $P$.

Question 3: specialization to location parameter estimation
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Same question with $g(Z;\theta) = Z - \theta$.

Observations
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My wild guesses are

- $\alpha_n$ converges in distribution to some chi-squared distribution.
- The solution of the problem will come from an application of the (functional) *delta method*.


  [1]: https://mathoverflow.net/q/313554/78539