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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?

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A very similar question was asked within the last week on MathOverflow. Multiple different approaches were given in the answers. – cardinal Feb 21 '12 at 18:30
up vote 2 down vote accepted

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).

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