Let $\{X_{n,i}:1\leq i \leq d_n\}$ be a triangular array of mean zero random variables where $d_n$ is a positive increasing sequence ($d_n\leq n$). Under some conditions, a Central Limit Theorem states that $$\frac{1}{\sqrt{d_n}}\sum_{i=1}^{d_n}X_{n,i}\overset{d}{\to} N\bigg(0,Var\bigg(\frac{1}{\sqrt{d_n}}\sum_{i=1}^{d_n}X_{n,i}\bigg)\bigg).$$
Problem
In my case, I'm working with the sum $\frac{1}{\sqrt{d_n}}\sum_{i\in J_n}X_{n,i}=\frac{1}{\sqrt{d_n}}\sum_{i=1}^nX_{n,i}1_{i\in J_n}$ where the cardinality of the set $J_n$ is $d_n$. $J_n$ is a set of consecutive indices. The difference from the above sum in the Theorem is that the smallest index of each $J_n$ does not need to be $1$. Also, for small $n$, say for all $n<n_0$, $J_n$ is empty, this means $d_n=0$. Although the sum I'm working on is not exactly the same of that of in the Theorem, it is intuitive that the weak convergence also holds. I believe this is a matter of defining the triangular array properly.
Consider the triangular array $\{X_{n,i}:i\in J_n, n\geq 1\}$. To apply the above theorem, I'm thinking of bring the indices in $J_n$ to the initial portion of each line of the triangle array. Like this:
My question is, if I show that $$\frac{1}{\sqrt{d_n}}\sum_{i=1}^{d_n}Z_{n,i}\overset{d}{\to} N\bigg(0,Var\bigg(\frac{1}{\sqrt{d_n}}\sum_{i=1}^{d_n}Z_{n,i}\bigg)\bigg),$$ does $$\frac{1}{\sqrt{d_n}}\sum_{i\in J_n}X_{n,i}\overset{d}{\to} N\bigg(0,Var\bigg(\frac{1}{\sqrt{d_n}}\sum_{i\in J_n}X_{n,i}\bigg)\bigg),$$ also holds?
*Note that $\sum_{i\in J_n}X_{n,i}=\sum_{i=1}^{d_n}Z_{n,i}$