I am trying to understand the concept of uniform convergence in probability. Suppose we have a sequence of random functions $M_n(\theta)$ that converge uniformly in probability to $0$,
$$ \sup\limits_{\theta \in \Theta}|M_n(\theta)| \xrightarrow{P} 0 $$
where $\Theta$ is a compact set. Then would the statement still hold if $\theta$ were not deterministic, but a random sequence $\theta_n \in \Theta$?
From a statistical point of view, this would be like proving consistency but with a plug-in estimator of $\theta$.
Yes, because, for every $\omega$ in the sample space, $|M_n(\theta (\omega))| \leq \sup_{\theta \in \Theta} |M_n(\theta)|$.