Let $F_{\theta}(x)$ denote a cumulative distribution function indexed by the parameter vector $\theta$. Given this definition is the following equation correct (and if so under which conditions)?
$$\lim_{\theta \rightarrow \theta^*}\frac{\partial}{\partial \theta} F_{\theta}(x) \stackrel{???}{=} \frac{\partial}{\partial \theta^*} \lim_{\theta \rightarrow \theta^*} F_{\theta}(x)$$
Thank you very much!
I believe you can exchange the limit and the derivative as long as your function is continuously differentiable in $\theta$ at $\theta^*$. That is, both your CDF and its corresponding PDF need to be continuous at whatever value of $\theta^*$ you are looking at. Assuming this is true, you will have: $$\lim_{\theta \rightarrow \theta^*}\frac{\partial}{\partial \theta} F_{\theta}(x) {=}\frac{\partial}{\partial \theta^*} F_{\theta^*}(x) $$ and $$\frac{\partial}{\partial \theta^*} \lim_{\theta \rightarrow \theta^*} F_{\theta}(x){=}\frac{\partial}{\partial \theta^*} F_{\theta^*}(x)$$ because by definition, $\lim_{x \rightarrow a}f(x){=}f(a)$ for a continuous function.
Note that whether or not the function is continuous in $x$ doesn't matter. For example, the $poisson(\lambda)$, which only has a single continuous parameter $\lambda$, would be one you could exchange the limit on for any valid value of $\lambda$ even though it is not continuous in $x$, but the $\chi^2(k)$ would not be interchangeable everywhere despite being a continuous distribution in $x$ since $k$ is a discrete, discontinuous parameter.
