I have already asked that Question on Cross Validated:
[Link][1]


  [1]: https://stats.stackexchange.com/questions/353956/how-to-find-the-optimal-convergence-rate?

Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference.
Assume that there is a asymptotical (weak) consistency result, i.e. there is an estimator $\theta_{n}(X_{1},\ldots,X_{2})$, s.t. 
$\theta_{n}(X_{1},\ldots,X_{2})\overset{\mathbb{P}}{\longrightarrow}\theta$.

Now, I want to go one step further, calculating a weak limit theorem allowing for testing and confidence on $\theta$. 
This means, I am looking for a deterministic sequence $\ell_{n}$, s.t.
$$\ell_{n}\cdot\left(\theta_{n}(X_{1},...,X_{n}\right)-\theta)\longrightarrow V$$ in distribution, with some r.v. $V$, s.t. $\mathbb{P}_{V}$ is a well known distribution. 

**Question 1:** 
Is there a way to find the *optimal* convergence rate $\ell_{n}$?

**Question 2:**
Is it possible to change the limit distribution from $V$ to $V'$ to get a different sequence $\ell_{n}'$? Of course, both distributions, $\mathbb{P}_{V}, \mathbb{P}_{V'}$ are assumed to be non-degenerated.