Let $\{w_i\}$ be a sequence of iid random variables, with $E[w_i]=0, V[w_i]=1$. Let $\{a_{i,N}\}_{i=1}^N$ be N real numbers that depend on N.
Let $s_N=\sqrt{\sum\limits_{i=1}^Na_{i,N}^2}$.
Is there any theory/books/proofs anywhere that deals with the convergence in distribution of
$\frac{1}{S_N}\sum\limits_{i=1}^Na_{i,N}w_i$?
The reason I am not able to use the ordinary central limit theorem, or Lindbergs CLT is that for each N the coefficients in front of $w_i$ changes.
Is there a standard way to deal with these sums? It seem likely that some work on this has been done, but I am not able to find it.
Please put your problems in the regime of triangle array of RVs,(c.f. P. Billingsley, Probability and Measures, Anniversary Ed., J. Wiley & Sons Inc.(2001), Sec27.6--), then you can verify that the Lindeberg condition is satisfied if the following limit holds $$ \lim_{n\to\infty}\dfrac{\max_{1\le i\le n}|a_{n,i}|}{s_n}=0. \tag{1} $$ Therefore, you could get CLT under (1).