TITLE: Weak convergence of a triangular array of Bernoulli-RV's
QUESTION [5 upvotes]: I am looking at the series $$X_{1,1},$$$$X_{2,1}, X_{2,2}$$ $$X_{3,1},X_{3,2},X_{3,3}$$ $$\dots$$ of independent r.v's with $p_n:=P(X_{n,k}=1)=n^{-\frac{1}{4}}$ and  $q_n:=P(X_{n,k}=0)=1-n^{-\frac{1}{4}}$. So they are Bernoulli-distributed.

I would like to know if$$S_n:=\frac{\sum_{k\leq n}(X_{n,k}-E(X_{n,k}))}{Var(\sum_{k\leq n}X_{n,k}) }$$ converges weakly, for $n\rightarrow \infty$.

One can observe that for every $n$ the sums  $\sum_{k\leq n}X_{n,k} (=:Y_n)$ are $B(n,p_n)$ distributed. One gets 

$E(X_{n,k})=p_n$,
$E(Y_n)=np_n$,
$Var(Y_n)=np_nq_n$.

So it is $$S_n=\frac{Y_n-np_n}{np_nq_n }$$.
The standard CLT can't be applied because the $Y_n$ have different winning-probabilities $p_n$. Also $Y_n$ does not converge to a Poisson-distributed r.v. because $$np_n=n^{\frac{3}{4}}$$ is not constant.
In which way can I apply the CLT?

REPLY [2 votes]: I assume your definition of $S_n$ wants a square root in the denominator; otherwise it converges to 0.
You want the Lindeberg-Feller central limit theorem.  See Theorem 3.4.5 of R. Durrett, Probability: Theory and Examples (4th edition).