I observe a sequence of r.v. $X_1, X_2, \dots$ where each $X_i$ is a function of the sample size $n$.
When $n \rightarrow \infty$ I have the following result: $X_1 \rightarrow^d E_1, X_2 \rightarrow^d E_2, \dots$ where $E_i$ are i.i.d random variable with mean $\mu$, variance $\sigma^2 < \infty$ and same density $f_E$. Moreover,
- $$\lim_{n \rightarrow \infty}E(X_i) = \mu$$
- $$\lim_{n \rightarrow \infty}V(X_i) = \sigma^2$$
- $$\textrm{Cov}(X_i, X_j) = O(n^{-1})$$
Denote $\bar X_n = n^{-1}\sum_{i = 1}^n X_i$, can I claim that (Linderberg-Levy CLT type)
$$ \sqrt{n}\left(\bar{X}_{n}-\mu\right) \stackrel{d}{\rightarrow} \mathcal{N}\left(0, \sigma^{2}\right) ? $$
In other words: if the dependence of the elements of the summation fades only asymptotically, does the CLT still holds?