Here's the question
Consider a sequence of estimators $X_{i}=X_{1},X_{2},...,X_{N}$ for $i=1,...,N$. Each estimator is resulting from a sample $j=1,...,n$. For each estimator $X_{i}$ the asymptotic normality is satisfied, as $n \rightarrow \infty$ with $N(0,X_{i}^2)$ and $σ^2=X_{i}^2$. Can we say that the asymptotic normality of the sequence $X_{i}$ is satisfied?