I have a sequence of continuous random variables $\{X_n\}$, with density $f_n(x \mid \theta)$ w.r.t. the Lebesgue measure. $X_n = o_{p_\theta}(1)$ for any $\theta \in \Theta$. For a fixed $\theta_0$ in the interior of the parameter space $\Theta$, let $B_\gamma(\theta_0)$ denote the closed ball of radius $\gamma$ centered at $\theta_0$.
I want to understand sufficient conditions for uniform convergence in probability in the following sense: there exists $\gamma>0$, such that for any $\epsilon>0$, $\delta>0$, there exists $N$ large, for all $n>N$,
$$\sup_{\theta \in B_\gamma(\theta_0)} Pr(|X_n| > \epsilon \mid \theta) < \delta.$$
What I have tried/ thought about so far:
I came across this paper which gives sufficient and necessary conditions for uniform convergence in probability. However, its setup is a bit different from mine. It discusses cases like: say $X_n(\theta)$'s are random functions of $\theta$, and then trying to see when $\sup_\theta X_n(\theta) = o_p(1)$. I've thought about it, but I don't see how I can transform my setup into such situation.
I also saw this post. The second equation listed there would be what I'm interested in. But the post didn't further discuss conditions for it to hold.
An equivalent question here is, I want to find a sequence of positive numbers $\epsilon_n \to 0$, and
$$\sup_{\theta \in B_\gamma(\theta_0)}Pr(|X_n| > \epsilon_n \mid \theta) \to 0.$$
- Another sufficient condition for this uniform convergence in probability is if the following holds:
$$\sup_{\theta \in B_\gamma(\theta_0)} f_n(x \mid \theta) \le \sum_{j=1}^K M_j f_n(x \mid \theta_j)$$
for some finite $K$, $M_j$'s in $\mathbb{R}$ and fixed $\theta_1, \dots, \theta_K$.
I'm interested to see whether there are other sufficient (or even better, sufficient and necessary) conditions for this uniform convergence in probability to hold. I've tried but didn't seem to find many relevant references. Any thoughts, comments, references would be greatly appreciated!!