Let $X_n$ be independent Bernoulli random variables with varying parameters $p_n$, that is, $P(X_n=1)=1-P(X_n=0)=p_n$. Given a sequence of real numbers $r_n$, how does one find, or at least hope to find, constants $c_n\in\mathbb R$ and $d_n>0$ so that $\dfrac{1}{d_n}(\sum_{k=1}^n r_kX_k-c_n)$ converges in distribution to a non-degenerate random variable?
Of course one way is to take $c_n=E(\sum_{k=1}^n r_kX_k)=\sum_{k=1}^n r_kp_k$ and $d_n^2=Var(\sum_{k=1}^n r_kX_k)=\sum_{k=1}^n r_k^2p_k(1-p_k)$. Then one can check if Lindeberg condition holds and if we are lucky we have CLT. What happens if Lindeberg condition does not hold? Then how do we guess what the normalization can be?
Note that I am not insisting that $d_n\to\infty$. Sometimes we may have that for a particular choice of $\{p_n\}$, $S_n$ converges a.s. without normalization. In that case, $d_n$ can be taken to be 1. The limit distribution will also not be degenerate. (The last fact follows from the fact that convergence in distribution of sum of independent r.v.'s implies almost sure convergence of the sum).
So I am looking for some general methods. Any reference will be welcome.