I am trying to apply a uniform law of large numbers, which is stated in Lemma 7.2 of "Econometrics" by Fumio Hayashi. The starting point is the stochastic process $\{x_t\}$, which we assume is stationary ergodic. The theorem then states (using different notation than me) that under some regularity conditions, the loss function
\begin{equation} L_T(\theta) = \frac{1}{T} \sum_{t=1}^T \ell(x_t; \theta) \end{equation}
converges in probability to $\mathbf{E}[\ell(x_t; \theta)]$, uniformly in $\theta$.
The problem, however, is that my loss function is on the form
\begin{equation} L_T(\theta) = \frac{1}{T} \sum_{t=1}^T \ell(x_1, \dots, x_t; \theta) \end{equation}
This loss arises from the log-likelihood of a sequence model: $\ell(x_1, \dots, x_t; \theta) = \log p_{\theta}(x_t \, | \, x_1, \dots, x_{t-1})$.
Do there exist similar results, or assumptions I can make to apply this result in my setting? I would guess that I need to assume that the effect of $x_{t-s}$ on $\ell(x_1, \dots, x_t; \theta)$ decays with $s$ in some way (maybe some kind of geometric bound on the covariance of $x_t$ could achieve this?), but I am not able to turn this into sufficiently precise statements as of yet.