I have already asked that Question on Cross Validated:
[Link](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-degenerate.



How to find the optimal convergence rate?
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\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\Var}{\operatorname{\mathsf Var}} 
\renewcommand{\P}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}} 
\newcommand{\eD}{\overset{\text{D}}\to} 
\newcommand{\D}{\overset{\text{D}}=} 
\newcommand{\tsi}{\tilde\si}$ 


To answer your Question 1, one needs to know what you mean by the optimal rate. 


However, as the following answer to your Question 2 shows, the convergence rate is essentially unique, up to the sign and rescaling, and so, it is hardly meaningful to talk about the optimal rate. Indeed, we have 



> 
> **Proposition.** Suppose that $(T\_n)$ is a sequence of random variables (r.v.'s) and $(\ell\_n)$ and $(\ell'\_n)$ are sequences of real numbers such that 
>  \begin{equation}
>  \ell\_n T\_n\eD V\quad\text{and}\quad\ell'\_n T\_n\eD V'
> \end{equation}
>  for some non-degenerate r.v.'s $V$ and $V'$, where $\eD$ denotes the convergence in distribution. Then for 
>  \begin{equation}
>  r\_n:=\frac{\ell'\_n}{\ell\_n}
> \end{equation}
>  we have $|r\_n|\to c$ for some nonzero real $c$. If, moreover, (the distribution of) $V$ is not symmetric, then $r\_n\to c$ for some nonzero real $c$. 
> 
> 
> 


*Proof.* Suppose that $r\_n\not\to c$ for any nonzero real $c$. Then (passing to subsequences if needed), we see that without loss of generality (wlog) one of the following three cases must obtain. 


*Case 1: $r\_n\to0$.* Then $\ell'\_n T\_n=r\_n(\ell\_n T\_n)\eD0V=0$, which contradicts the conditions that $\ell'\_n T\_n\eD V'$ and $V'$ is non-degenerate. 


*Case 2: $|r\_n|\to\infty$.* This reduces to Case 1 by switching the roles of $\ell\_n$ and $V$ with those of $\ell'\_n$ and $V'$. 


*Case 3: $r\_{m\_k}\to r$ and $r\_{n\_k}\to s$, where $r$ and $s$ are distinct nonzero real numbers and $(m\_k)$ and $(n\_k)$ are strictly increasing sequences of natural numbers.* Then 
$$\ell'\_{m\_k} T\_{m\_k}=r\_{m\_k}(\ell\_{m\_k} T\_{m\_k})\eD rV=:W$$ 
and similarly
$$\ell'\_{n\_k} T\_{n\_k}\eD sV=\ga W,$$
where $\ga:=s/r$. In view of the condition $\ell'\_n T\_n\eD V'$, we have 
$$W\D\ga W,$$
where $\D$ means the equality in distribution. Wlog $|\ga|\le1$ and hence one of the following two subcases of Case 3 must obtain. 


*Subcase 3.1: $|\ga|=1$.* Then
$\ga=-1$, because $\ga=s/r$ and $r$ and $s$ are distinct. So, in this subcase $W\D-W$, that is, $W$ is symmetric and hence $V$ is so. 


*Subcase 3.2: $|\ga|<1$.* Then $W\D\ga W\D\ga^2 W\D\dots\D\ga^n W\eD0$, so that $W\D0$ and hence $V\D0$, a contradiction. 


Thus, one may have $r\_n\not\to c$ for any nonzero real $c$ only if $V$ is symmetric. 


Similarly, by applying the absolute value function to $\ell\_n$, $\ell'\_n$, $r\_n$, $T\_n$, $V$, and $V'$, it is shown that $|r\_n|\to c$ for some nonzero real $c$ in all cases without exceptions. (In this latter consideration, there will be no analogue of the exceptional Subcase~3.1.) The proposition is now proved. $\Box$


