Is there a Central limit theorem in which the variance of the random variables depends on the sample size? For example, assume $X_1$, ..., $X_n$, $n$ independent random variables such that $E(X_i) = \mu_n$ and $V(X_i) = \sigma^2_n$, for any $i = 1$, $\dots$, $n$.
I assume that $\lim_{n\to\infty} \mu_n = \mu$ and $\lim_{n\to\infty} \sigma^2_n = \sigma^2$.
What is the convergence in law of $\sqrt{n}(\bar{X}_n - \mu)$, where $\bar{x}_n = \dfrac{1}{n}\sum_{i}^n X_i$?
Can I say that $\sqrt{n}(\bar{X}_n - \mu) \overset{d}{\to} N(0, \sigma^2)$? If yes, could you please give a reference? This is actually slightly different from the standard central limit theorem I now, in which the mean and the variance do not depend on $n$.
An example of random variables is, $X_i = (1 + \frac{1}{n})Y_i$ where $Y_i$'s are i.i.d.
EDIT: For the example $X_i = (1 + \frac{1}{n})Y_i$, I can use the standard CLT on $Y$ and then use Slutsky's lemma. But assume that I want to establish the convergence directly using a CLT. Or assume that the function is a bit more complicated than a simple product.