Is the delta method valid also for non-normal variables?
Claim:
Let $\sqrt{n}(\hat{X}_n-\theta) \xrightarrow{d} \hat{f} $.
With $\hat f$ having a finite distribution.
Then for every $g$ such that $g'(\theta)$ is nonzero we have:
$\sqrt{n}(g(\hat{X}_n)-g(\theta)) \xrightarrow{d} g'(\theta)\hat{f} $.
It seems to me that the standard proof:
https://en.wikipedia.org/wiki/Delta_method
applies also here, the only point that we need to assess that $\hat{X}_n \xrightarrow{P} \theta$.
I have also the impression that also $\sqrt{n}$ can be substituted by any increasing function of $n$.
Am I thinking wrong? Where can I find such generalizations of the delta method?
