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?