# Asymptotic normality of the maximum likelihood estimator with dependent data

In the setup, assume $$\left(\mathbb{R}, \mathscr{B}\left(\mathbb{R}\right), P\right)$$ is the underlying probability space and suppose that $$\left\{\mathcal{F_n}\right\}_{n\in \mathbb{N}}$$ is a filtration inside the Borel sets.

I am interested in recovering the asymptotic normality of a maximum likelihood estimator with dependent data using a martingale central limit theorem. Let $$\left\{P_\theta\right\}_{\theta \in \Theta}$$ be a family of mutually absolutely continuous measures on $$\mathscr{B}\left(\mathbb{R}\right)$$ where $$\Theta \subset \mathbb{R}$$ is an open interval. Further assume that $$P_\theta \ll P$$ for all $$\theta \in \Theta$$. Define $$l_n\left(\theta\right) := \log\left(\mathbb{E}_P\left[\frac{dP_\theta}{dP} \mid \mathcal{F}_n\right]\right)$$ as the log-likelihood function where the likelihood is a density of the restriction of the measures $$P_\theta,P$$ to the sub-sigma algebra $$\mathcal{F}_n$$. Define the maximum likelihood estimator $$\hat{\theta}_n$$ as a solution to $$l_{n}^\prime\left(\theta\right) = 0$$, assuming that the solution exists and is unique. Suppose further that the quadratic variation $$\frac{\left[l^\prime(\theta), l^\prime(\theta)\right]_n }{n} \stackrel{p}{\to} \sigma^2$$ where $$\sigma > 0$$. What assumptions and what CLT do we need to ensure that $$\sqrt{n}\left(\hat{\theta}_n - \theta \right) \to \mathcal{N}\left(0,\sigma^{-2}\right)$$ and $$-\frac{l^{\prime\prime}_n\left(\hat{\theta}_n\right)}{n}\to \sigma^2 ?$$