Central Limit Theorem with (linear) weights

Question

I have a question about the CLT. Suppose we have the independently and identically distributed random variables $X_i$ with mean $\mu$ and variance $\sigma^2$. Then, by the Central Limit Theorem

$$ \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i-\mu\right) \rightarrow N\left(0, \sigma^2\right)\Longleftrightarrow \sum_{i=1}^n X_i \rightarrow N(n\mu, n\sigma^2), $$ as $n\rightarrow \infty$. But what if we have $\sum_{i=1}^nc_iX_i$, where $c_i$ are constants? Is it simply to say that

$$ \sum_{i=1}^nc_iX_i \rightarrow N\left(E\left[\sum_{i=1}^nc_iX_i\right], \;V\left[\sum_{i=1}^nc_iX_i\right]\right), $$

that is,

$$ \sum_{i=1}^nc_iX_i \rightarrow N\left(\mu\sum_{i=1}^nc_i,\; \sigma^2\sum_{i=1}^nc_i^2\right), $$

or am I missing something here?

Answer 1

You might be after central limit theorems à la Lindeberg or Lyapunov, see here. These yield the convergence you are interested in, but stated more correctly, for example, some conditions ensuring that the sequence of general term $$ \frac{\sum\limits_{i=1}^nc_iX_i-\mu\sum\limits_{i=1}^nc_i}{\sigma\cdot\sqrt{\sum\limits_{i=1}^nc_i^2}} $$ converges in distribution to a standard normal distribution.